My first thought is "heat related."
Is the computer really shutting all the way off, or is the video going blank and never coming back (but the power light is still on)? If the power light is still on, suspect video. When video cards overheat, they usually shut down the video processor for safety, but the CPU is still running.
If it's CPU related, the whole machine might shut down or just freeze up. (Depends on which motherboard we're talking about.) Make sure that the heatsink is still properly attached tightly (and oriented correctly and all that) and the fan has power/isn't blocked/is spinning.
Might be memory, but I'm still leaning towards the heat angle. Download and run Memtest86 and give it a few tries. That might help narrow it down to CPU heat as well. Memtest86 doesn't have high video requirements, so any video card would be fine, but it really exercises the CPU.
Case: Enermax CS-A106USB
P/S: OCZ OCZ700GXSSLI 700W
Motherboard: ASUS P5N32-E SLI
CPU: Intel E6300(idle 35C/load 39C)
RAM: OCZ Platinum Rev 2 DDR2 800 MHz (4 x 1GB) 4-4-4-15-T1 (2.1v)
Hard Disk: Seagate 2x250GB SATA ST3250620AS in RAID1
Monitor: Mitsubishi DP900u
Graphics: BFG GeForce 8800GT PCI-E
Sound: SoundBlaster X-Fi Fatal1ty
Speakers: Klipsch ProMedia v.2-400 4.1 speakers
Optical: LiteOn 16X DVD-ROM; Sony DRU-710A DVD-RW