So, the next question is whether:
(a) Play safe and go 2 seperate rigs with 3 cards each....OR
(b) Attempt to be a tech ninja and run all 6 off one mobo....
B isn't even an option, so you'll have to go with A. The 5970 is a dual-gpu card, so each card shows up as two GPUs to the OS. You can't have 12 GPUs, as even in a 64-bit OS the drivers cap out at 8. Twin rigs with 3 each are fine, as long as your OS is 64-bit, as that's only 6 GPUs per rig.
TL;DR
5970 = Dual GPU card
32-bit OS = limited to 4 GPUs (or 2 dual-GPU cards)
64-bit OS = limited to 8 GPUs (or 4 dual-GPU cards)