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I'd like to see if this is a reasonable solution, or is just plain terrible.

I wouldn't say it isn't "terrible" - mostly because it works and doesn't appear to be very inefficient. There are some improvements that can be made.

• use strict comparison - i.e. === when comparing values
• Instead of setting the value in each case, append to output, which allows one to eliminate the first conditional, as well as the else keywords and then if output is still an empty string, set it to i using short-circuiting logic. That way there will only be two modulus division operations instead of three.
• Use consistent indentation for readability. Maybe it was the case that your code was indented properly but when you pasted here it was thrown off because of the markdown formatting...

for (var i = 1; i <= 100; i++) {
    var output = "";

    if (i % 3 === 0) {
        output += "Fizz";
    }
    if (i % 5 === 0) {
        output += "Buzz";
    }
    output = output || i;

    console.log(output);
}

Try comparison in this jsperf.