Similar to Bob's description I imagine, I made drying screens from four lengths of wood, some flyscreen and a staple gun. These lie horizontally in an old spare cupboard. I then lie washed prints face down to dry for 24-48 hours.
After this they are reasonably flat, certainly flat enough to put into my dry mount press - or you could just put them in a book, use some release paper to make a sandwich if worried about the book print marking them and put a weight on top.
I don't experience any wrinkling from the press but I only print up to 8x10" and I always put prints in a book for further flattening for a week or two following their 30 seconds in the press.
It's certainly a slightly frustrating step in the process but not as frustrating as the next step - mounting and framing! These are good frustrations to have though, for an amateur like myself at least.