Yep, +1 on the Baltic Birch. I did the same thing with a dresser that I've had since I was a kid--I built new drawers and remove the old drawer fronts to re-use. I used the "cabinet grade" plywood and drawer slides at Home Depot. What a waste of money and time that was! The drawers warped just slightly, but enough to bind up the drawer slides (which are cheap copies of Accuride slides--next time I'll go with Accuride or Blum and skip the cheap Chinese imports.)
tl;dr: don't skimp on materials--get genuine Baltic Birch plywood and name-brand slides.