My newly built PC keeps crashing with this screen:
 

 
I cannot figure out what the problem is as I do not have any compatible parts to swap and test.
 
The crashing occurs randomly.  Sometimes 10 seconds within Windows booting, sometimes hours.  However, as soon as I load a game in Crysis the system crashes within 15 seconds.  I already have RMA'd the motherboard for another of the same model and still have the same problem.  Also, using the onboard video instead does not resolve the problem.
 
I have run countless stress/torture tests and benchmarks and all the results are ~normal, aside from the slightly-lower-than-expected scores on the benchmarks (in no particular area).
 
 
Specs:
 
ECS A780GM-A AMD780G Socket AM2+ Motherboard
 
AMD Phenom Quad 9500+ 2.2Ghz Socket AM2+ CPU
 
G.SKILL 4GB (2 x 2GB) 240-Pin DDR2 SDRAM DDR2 1066 (PC2 8500)
 
SAPPHIRE Radeon HD 3870 512MB 256-bit GDDR4 PCI Express 2.0 x16 HDCP  Video Card
 
OCZ StealthXStream OCZ500SXS ATX12V / EPS12V 500W Power Supply
 
 
 
What is the next step I should take it fixing this system?

Any help?

Could be drivers, power supply, or memory.  Have you run memtest86 yet?  Have you rebuilt your OS after reformatting your drive?

Is this a crash or just the monitor loosing the picture? Have you tried a new monitor? Maybe your monitor has a prob with the frequency output range of 3870. You can also adjust the 3D refresh rate overide from within the Catalyst driver if you are using an older CRT monitor.  
 
Crysis has some issues when played at a resolution different from that of the desktop. Although they were not what you're experiencing, but you never know. You could try this and see if it fixes the problem. If you play Crysis at a lower res than the desktop set the desktop back to the res at which you play Crysis. Play the game and manually reset the desktop once you have finished.  
 
If your mobo has an ATi graphics chip on it, which I believe 780G's do, but I don't own one, then I'd also try disabling that device in the BIOS in case you have some driver contamination there. I'd also remove all ATi Drivers reboot and reinstall after the onboard GPU is disabled.
 
Good Luck