Scott, suggested that maybe strings would be simpler than rack/pinion and threaded rods. I'm considering all options. Thanks Scott!

I have not figure out how to apply the string and pulley concept to pick up and move the chess pieces. I thought I could use threaded rod to skewer a case that holds each chessmen, but then I realized that to actually pull it off the shelf, I need yet another mechanism to slide the box in the Z direction. Grr!

Scott's video: