Well...I got the replacement motherboard, but...things still aren't quite right.
This time it boots up into BIOS with no problems (with video output too!), but only if I don't have the sticks of RAM in slots 1 or 3.
I can boot with a single stick in either slot 2 or 4, or with both sticks in those slots, but if I put a stick (either of them) in either slot 1 or 3, it tries to boot for about 15 seconds (with no video output) and then restarts.
I've tried each stick in every slot, and every possible combination of both.