The crappy drawers in my kitchen were falling apart, so I took the faceplaces off and built new drawers. I used 3/4" melamine-backed plywood for the sides, and 1/4" melamine-backed MDF for the bottom. You can see a notch in the lowest tenon of each box joint where I cut the dado for the bottom.
Everything's glued and nailed in place. The nails through the box joints are another cosmetic issue, but I wanted the strength. I guess if I cared enough, I could fill and paint it all smooth.