Troubleshooting a system - Board or Power Supply?

Last fall I built a variation of the June 2011 Budget system, though I upgraded the motherboard and video card slightly. For the board I switched to a ASRock H61M/U3S3 to get SATA 3, and all worked well for the first several months. Around that time I started getting occasional reboots and blue screens. The stop codes were pointing to different things, but I never gave it much thought since the problem was rare. I should add that the reboots and blue screens only happened when the system was idle, and never once under load. A couple of weeks ago the frequency of incidents increased dramatically, with a dead machine after another 48hrs.

At first the stop codes had me thinking that I might have a bad hard drive. I tried a clean windows install on a new drive, but I would get blue screens and an assortment of memory related codes during the install. Accordingly I started testing the RAM, and I found that the issue appears to be tied to this part of the system. The machine will run fine on either 4GB memory chip and it does not matter which of the two slots I have the memory in, but if I try to run the machine on both 4GB memory chips, blue screens return.

My assumption is that I either have a bad board or bad power supply. Is that reasonable thinking? If so, could anyone offer any thoughts on which to try to replace first? If my assumption sounds wrong, could anyone point me in the right direction? Is there any chance that if one of the memory chips is bad that it would cause a conflict only when a second chip is present?

Any guidance would be greatly appreciated, thanks.