There have been several discussion on this forum were people sometimes have issues between the PSU and their Intel motherboard. Using a different model PSU often solves the problem.
For the DP67BG there has been an enhancement to the Power Supply Circuitry.
AA code of new boards is G10491-305 and higher. Google for Product Change Notification 110583 - 00 for details.
Thanks for the information, vbaderks.
I eventually went to the expense of buying a new mainboard, the Intel DX79SR Extreme Desktop Board. I assembled my system + the Corsair AX1200i two days ago, and so far everything's been working like a charm. I'm not having any issues at all, neither during booting up nor during operation.
It really seems that the AX1200i as well as the replacement unit I got didn't like my 'old' Intel
DP67BG (or vice versa).
Problem solved for me, albeit at a high price.