I just ran memtest for about 24 hours, and it appers to have failed miserably. After the first 8 hours or so I had 1 error in the "red" part of the display. Now, after the temperature in the room has risen a bit, I have the whole red part filled with errors just from one test from one pass. Not sure if I can scroll this text in any way, but I'm guessing it's pretty bad.
The summary part of the memtest says I have had over 1200 errors in 26 passes.
I'm no expert on reading memtest results, but I guess it's safe to assume that this is bad. I'll have to pick up some new RAM to test if the system will run fine with the 5970.