In the past I have used just construction foam, but gluing the construction foam onto some plywood or plastic boards so you can add furniture feet and have clearance for wiring underneath if you need it.
Liquid nails holds foam to just about anything.
Small screws from the bottom of the board hold pieces on well.
I've also seen people get great results using corkboard as Diorama bases. That'll probably be the next thing I try for a scratch built one.