Instead of soldering up a bunch of resistors, and shift registers, I did a little googling instead.
Check out what I found.
http://aglick.com/charliecube.html
This guy built a 4x4x4 RGB cube using 64 LEDs, some wire, perf board, and a nano. No resistors or transistors or chips of any kind, just some wire, leds, board, and nano.
The way he solved the problem was like using geometry to solve a accounting problem.