I re-uploaded a new sketch, and hooked it up again, and its doing something! 
Right now I am only controlling one plane by connecting it to 5v.
The ICs are wired up like the example in the TLC5940 tutorial:
http://playground.arduino.cc/Learning/TLC5940
except that I have decoupling caps on each chip.
Its running the Useprogmem example program, it just lights all 16 RGB LEDs in a plane, and turns them off in sequence. I think it may do some PWM while its turning it off too.
Its not as bright as I had hoped, but maybe with some experimentation, i can make some improvements. I ran the same LEDs on shift registers and resistors, and I seem to remember them being considerably brighter, but maybe that just my imagination.