The Rainbow Cube schematic is for 64 RGB LEDs that are arranged on the really unusual thin PCBs that are both structural and wiring elements for the cube, as well as the “panel”, which is a base PCB that has about 5 resistors, a voltage regulator, the interface socket pins that plug into the Rainbowduino, and a few other parts, but no other chips. Aside from the LED arrangement, it is fairly simple.
Without having studied the schematic in much detail, I assume you will need to create simple electronics with resistors for current limiting that goes between the LEDs and the interface pins.
I’m sorry that I have not studied it enough to tell you more, but I do hope that you will share what you learn - I might well like to build one of these myself.