OK, I've narrowed it down a bit more...
When the 4/8 pin connectors are NOT in place, when I hit the power switch, the fans all spin and keep spinning, but that's all.
When it IS connected, the fans spin for a fraction of a second and then stop, with nothing else happening.
If anyone else has had this problem with the board (now I'm pretty sure it's the board), please let me know, otherwise I'm going to fill out an RMA tomorrow for the PSU and board.