As @maxschlepzig said in a comment,

Note that there is the MYSQL_PWD environment variable which is read by mysql if you don't specify -p.

So in bash:

read -s -p "Enter the mysql password for $DBUSER @ $DBNAME: " DBPASS 
export MYSQL_PWD="$DBPASS" 
mysqldump -u $DBUSER $DBNAME > dump.sql

As @maxschlepzig said in a comment,

Note that there is the MYSQL_PWD environment variable which is read by mysql if you don't specify -p.

So in bash:

read -s -p "Enter the mysql password for $DBUSER @ $DBNAME: " DBPASS 
export MYSQL_PWD="$DBPASS" 
mysqldump -u $DBUSER $DBNAME > dump.sql 

# once you are done
export MYSQL_PWD=""