Use zippered pencil pouches to organize puzzles instead of storing the bulky boxes. Cut off the image from the front of the box and store it in the bag, too. (This blog post has several great ideas for organizing kids toys.)
I am absolutely building these for the empty wall in my kitchen...they'll be on either side of my basement door. Except...I'm going to stain the pieces and use jewel tone versions of the game colors. Plus, I'm adding doors to keep the contents dust-free.