Hi,
I got my cabinets with no particle board, birch plywood sides, solid maple shelves and drawer boxes. The interiors are factory-finished with a catalyzed lacquer, and the hardwood external parts I finished myself. The best part is, there is no detectable odor from them. I didn't do this to avoid formaldehyde, per se, but my plan to minimize man-made materials essentially accomplished the same thing. Coating offending materials with shellac is a fairly effective odor barrier.
Casey

I have made cabinets from formaldehyde-free melamine. It was about $10 a sheet more than regular, worked fine. I prefer the pre-finished plywood because it's better looking & lighter. My supplier doesn't bill it as formaldehyde-free, and there may be some in the glue, but it doesn't stink.