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function [found] = makeGrids(found,grid,depth,x,y)
    if (depth > (x*y))
        %If at max depth then grid is valid.
        found = [found grid]; %So add to list
        disp(grid)
        %found=found+1; %So one more found
    else
        %Otherwise another layer to do
        for k=1:depth
            %For each number in this layer
            grid(depth)=k; %Update grid with new number. Depth is linear index
            %Now we check to see if the current state of the grid is
            %acceptable (if it isn't then no lower down ones possibly can)
            if (checkPanels(grid,depth,x,y)) %If it is acceptable (i.e. there are no remaining values with no frame)
                found=makeGrids(found,grid,depth+1,x,y); %Work through the next layer
            end
        end
    end
end

function success = checkPanels(grid,depth,x,y)
    success = false;
    for ys=1:y
        for xs=1:x
            %Start at (1,1) and work through to (n,n)
            expected = xs+(ys-1)*x; %Linear index of this cell
            if(expected > depth)
                %If the expected val is more than current depth
                return; %Then we are done searching
            end
            panelFound=false;
            for xe=x:-1:xs
                for ye=y:-1:ys
                    %For each end value starting from largest to smallest
                    panel=grid(xs:xe,ys:ye); %Extract the current panel
                    panel(panel==expected)=0; %Cover all instances of expected value in panel
                    panelFound = all(all(~panel));%Check if every number in this panel is covered
                    if (panelFound) %If we have found a complete panel
                        grid(xs:xe,ys:ye) = -1; %then mark panel a covered in the grid.
                        break; %We can only have one panel for any given number, so break.
                    end
                end
                if (panelFound)
                    break; %We can only have one panel for any given number, so continue break.
                end
            end
            %Check if entire grid is covered
            if (all(all(grid==-1)))
                success = true; %Grid is all covered and valid
                return;
            end
        end
    end
end

%Grid Size
x=3; %3x3
y=3;
%Enumerate all grids
grids=makeGrids({},zeros(x,y),1,x,y);
gridCount = numel(grids);
disp(['Grids Found: ' num2str(gridCount)]);
%Colour mapping for image - hot(x*y), hsv(x*y) and jet(x*y) all look good
map=[jet(x*y);0,0,0]; %;0,0,0 is for black borderd
%Create images for all grids
for i=1:gridCount
    img=grids{i};
    img(x+1,:)=10;
    img(:,y+1)=10;
    [img, newmap]=imresize(img,map,32,'nearest');
    imwrite(ind2rgb(img,newmap),['Grid' num2str(i,'%03d') '.png']);
end

%Create tiled montage of images
dirOutput = dir('Grid*.png');
fileNames = {dirOutput.name}';
vertical = max(factor(gridCount));
horizontal = gridCount/vertical;
montage(fileNames, 'Size', [vertical horizontal]);
%And save montage as PNG.
allGrids=getframe(gca);
imwrite(allGrids.cdata,'AllGrids.png');

It's quite impressive how fast the number of possible grids grow.

For a 3x3 grid we have the 322 possibilities as was identified by the other answers. Jumping up to 4x4 gives 70878 possible grids. Going for a two page spread of 3x6 the number increases to a barmy 314662 possible grids!

function [found] = makeGrids(found,grid,depth,x,y)
    if (depth > (x*y))
        %If at max depth then grid is valid.
        found = [found grid]; %So add to list
        disp(grid)
        %found=found+1; %So one more found
    else
        %Show current depth and found count during search.
        if (depth<=(x*(y-1)))
            disp([repmat('..> ',1,depth) num2str(depth) ' (found:' num2str(numel(found)) ')']);
        end
        %Another layer to do
        for k=1:depth
            %For each number in this layer
            grid(depth)=k; %Update grid with new number. Depth is linear index
            %Now we check to see if the current state of the grid is
            %acceptable (if it isn't then no lower down ones possibly can)
            if (checkPanels(grid,depth,x,y)) %If it is acceptable (i.e. there are no remaining values with no frame)
                found=makeGrids(found,grid,depth+1,x,y); %Work through the next layer
            end
        end
    end
end

function success = checkPanels(grid,depth,x,y)
    success = false;
    for ys=1:y
        for xs=1:x
            %Start at (1,1) and work through to (n,n)
            expected = xs+(ys-1)*x; %Linear index of this cell
            if(expected > depth)
                %If the expected val is more than current depth
                return; %Then we are done searching
            end
            panelFound=false;
            for xe=x:-1:xs
                for ye=y:-1:ys
                    %For each end value starting from largest to smallest
                    panel=grid(xs:xe,ys:ye); %Extract the current panel
                    panel(panel==expected)=0; %Cover all instances of expected value in panel
                    panelFound = all(all(~panel));%Check if every number in this panel is covered
                    if (panelFound) %If we have found a complete panel
                        grid(xs:xe,ys:ye) = -1; %then mark panel a covered in the grid.
                        break; %We can only have one panel for any given number, so break.
                    end
                end
                if (panelFound)
                    break; %We can only have one panel for any given number, so continue break.
                end
            end
            %Check if entire grid is covered
            if (all(all(grid==-1)))
                success = true; %Grid is all covered and valid
                return;
            end
        end
    end
end

%Grid Size
x=3; %3x3
y=3;
%Enumerate all grids
grids=makeGrids({},zeros(x,y),1,x,y);
gridCount = numel(grids);
disp(['Grids Found: ' num2str(gridCount)]);
%Colour mapping for image - hot(x*y), hsv(x*y) and jet(x*y) all look good
map=[jet(x*y);0,0,0]; %;0,0,0 is for black borderd
%Create images for all grids
nameFormat = ['%0' num2str(ceil(log10(gridCount))) 'd'];
for i=1:gridCount
    img=grids{i};
    img(x+1,:)=(x*y)+1;
    img(:,y+1)=(x*y)+1;
    [img, newmap]=imresize(img,map,32,'nearest');
    imwrite(ind2rgb(img,newmap),['Grid' num2str(i,nameFormat) '.png']);
end

%Create tiled montage of images
dirOutput = dir('Grid*.png');
fileNames = {dirOutput.name}';
vertical = max(factor(gridCount));
horizontal = gridCount/vertical;
montage(fileNames, 'Size', [vertical horizontal]);
%And save montage as PNG.
allGrids=getframe(gca);
imwrite(allGrids.cdata,'AllGrids.png');

function [found] = makeGrids(found,grid,depth)
    if (depth == 10)
        %If at max depth then grid is valid.
        found = [found grid]; %So add to list
        disp(grid)
        %found=found+1; %So one more found
    else
        %Otherwise another layer to do
        if(depth==9)
            %BP
        end
        for k=1:depth
            %For each number in this layer
            grid(depth)=k; %Update grid with new number. Depth is linear index
            %Now we check to see if the current state of the grid is
            %acceptable (if it isn't then no lower down ones possibly can)
            if (checkPanels(grid,depth)) %If it is acceptable (i.e. there are no remaining values with no frame)
                found=makeGrids(found,grid,depth+1); %Work through the next layer
            end
        end
    end
end

function success = checkPanels(grid,depth)
    success = false;
    for ys=1:3
        for xs=1:3
            %Start at (1,1) and work through to (3,3)
            expected = xs+(ys-1)*3; %Linear index of this cell
            if(expected > depth)
                %If the expected val is more than current depth
                return; %Then we are done searching
            end
            panelFound=false;
            for xe=3:-1:xs
                for ye=3:-1:ys
                    %For each end value starting from largest to smallest
                    panel=grid(xs:xe,ys:ye); %Extract the current panel
                    panel(panel==expected)=0; %Cover all instances of expected value in panel
                    panelFound = all(all(~panel));%Check if every number in this panel is covered
                    if (panelFound) %If we have found a complete panel
                        grid(xs:xe,ys:ye) = -1; %then mark panel a covered in the grid.
                        break; %We can only have one panel for any given number, so break.
                    end
                end
                if (panelFound)
                    break; %We can only have one panel for any given number, so continue break.
                end
            end
            %Check if entire grid is covered
            if (all(all(grid==-1)))
                success = true; %Grid is all covered and valid
                return;
            end
        end
    end
end

%Enumerate all grids
grids=makeGrids({},zeros(3),1);
%Colour mapping for image - hot(9), hsv(9) and jet(9) all look good
map=[jet(9);0,0,0]; %;0,0,0 is for black borderd
%Create images for all grids
for i=1:numel(grids)
    img=grids{i};
    %Add border to bottom right
    img(4,:)=10;
    img(:,4)=10;
    %Resize to 32x32 per square
    [img, newmap]=imresize(img,map,32,'nearest');
    %Save as PNG
    imwrite(ind2rgb(img,newmap),['Grid' num2str(i,'%03d') '.png']);
end

%Create tiled montage of images
dirOutput = dir('Grid*.png');
fileNames = {dirOutput.name}';
montage(fileNames, 'Size', [23 14]);
%And save montage as PNG.
allGrids=getframe(gca);
imwrite(allGrids.cdata,'AllGrids.png');

The following MATLAB functions are used to produce the grids.

function [found] = makeGrids(found,grid,depth,x,y)
    if (depth > (x*y))
        %If at max depth then grid is valid.
        found = [found grid]; %So add to list
        disp(grid)
        %found=found+1; %So one more found
    else
        %Otherwise another layer to do
        for k=1:depth
            %For each number in this layer
            grid(depth)=k; %Update grid with new number. Depth is linear index
            %Now we check to see if the current state of the grid is
            %acceptable (if it isn't then no lower down ones possibly can)
            if (checkPanels(grid,depth,x,y)) %If it is acceptable (i.e. there are no remaining values with no frame)
                found=makeGrids(found,grid,depth+1,x,y); %Work through the next layer
            end
        end
    end
end

function success = checkPanels(grid,depth,x,y)
    success = false;
    for ys=1:y
        for xs=1:x
            %Start at (1,1) and work through to (n,n)
            expected = xs+(ys-1)*x; %Linear index of this cell
            if(expected > depth)
                %If the expected val is more than current depth
                return; %Then we are done searching
            end
            panelFound=false;
            for xe=x:-1:xs
                for ye=y:-1:ys
                    %For each end value starting from largest to smallest
                    panel=grid(xs:xe,ys:ye); %Extract the current panel
                    panel(panel==expected)=0; %Cover all instances of expected value in panel
                    panelFound = all(all(~panel));%Check if every number in this panel is covered
                    if (panelFound) %If we have found a complete panel
                        grid(xs:xe,ys:ye) = -1; %then mark panel a covered in the grid.
                        break; %We can only have one panel for any given number, so break.
                    end
                end
                if (panelFound)
                    break; %We can only have one panel for any given number, so continue break.
                end
            end
            %Check if entire grid is covered
            if (all(all(grid==-1)))
                success = true; %Grid is all covered and valid
                return;
            end
        end
    end
end

%Grid Size
x=3; %3x3
y=3;
%Enumerate all grids
grids=makeGrids({},zeros(x,y),1,x,y);
gridCount = numel(grids);
disp(['Grids Found: ' num2str(gridCount)]);
%Colour mapping for image - hot(x*y), hsv(x*y) and jet(x*y) all look good
map=[jet(x*y);0,0,0]; %;0,0,0 is for black borderd
%Create images for all grids
for i=1:gridCount
    img=grids{i};
    img(x+1,:)=10;
    img(:,y+1)=10;
    [img, newmap]=imresize(img,map,32,'nearest');
    imwrite(ind2rgb(img,newmap),['Grid' num2str(i,'%03d') '.png']);
end

%Create tiled montage of images
dirOutput = dir('Grid*.png');
fileNames = {dirOutput.name}';
vertical = max(factor(gridCount));
horizontal = gridCount/vertical;
montage(fileNames, 'Size', [vertical horizontal]);
%And save montage as PNG.
allGrids=getframe(gca);
imwrite(allGrids.cdata,'AllGrids.png');