How to hang 1/4" Plywood
Glue a strip (or two strips, to keep it parallel to the wall surface) of thicker wood to the back. 

Properly shaped (see ASCII art below for one version), that can BE your hanging hardware, or you can attach your hanging hardware to it.

X
X||   X is the plywood. This angle-cut bit of ||
X\|     wood is attached to the plywood.      \|
X\\                                            \
X|\
X||   X is the plywood. This angle-cut bit of \
X       wood is attached to the wall.         |\
X                                             ||     


"Swiss Clips" would another approach, with no glue or nails, but some hardware showing on the front side.