The best thing since slic3d bread! Modular 3D-printable DIY breadboard. For the connections underneath, I tested out using Aluminum Foil (I used Heavy Duty, just in case) and it seems to do the trick. I used a breadboard I had lying around to test the foil. I took a long strip and folded it up into nearly the shape of the cavity, and cut it with scissors to size, and it made the connection properly. Breadboards of this size aren't that expensive (~$10 at RadioShack), but I did some math on the cost of the raw materials (both plastic and aluminum foil). And the cost of 1 unit with a main connection board and 2 supply boards came out to be less than $2.

Quick Disclaimer: I have not tested how these parts print. The Breadboard Tops are supposed to be printed with the lettering down, so I'm not sure how the writing will come out, but the rest should be printing friendly. Once I get my printer running I will test this out myself, and post pictures and/or a video of the finished product. If you end up making it, let me know how it turns out!

EDIT-2021: Sorry for the issue in the original design: the pitch between successive holes should be 2.54mm (0.10") not 2.5mm.