Troubleshooting P8Z68 DELUXE/GEN3 Build

I'm trying to pin down a problem with my P8Z68 DELUXE/GEN3 system that
I built around two and a half years ago. The problem is that the
system will suddenly shut off and reboot, usually when the GPU is
pulling extra power (during gaming (Diablo 3), running Windows
Experience Index refresh, or when initiating a GPU stress test with
FurMark). The system will shut off completely (power light on case
goes off), and then will reboot by itself several seconds later.

I've run MemTest86+ and Prime95 extensively which would seem to rule
out the RAM and CPU. I have gone into the case and reseated all
cables, including the modular cable on the power supply. I've run
thorough OEM diagnostics on the SSD and HDDs. Power supply voltages in
BIOS look normal. I've even tried reverting to the previous NVIDIA
driver with no luck.

With the current GPU (MSI GTX 760), it will occasionally shut off when
I click the Burn-In Test button in FurMark. If it doesn't crash
initially, then it will run FurMark ok with the GTX 760. If I replace
the GTX 760 with the card I originally built the system with (an EVGA
GTX 580), it will shut off every time I attempt to run the Burn-In
Test in FurMark. Obviously the GTX 580 is a much more power hungry
card than the GTX 760, so it's looking like current draw has something
to do with the problem.

This was a totally stable system until just a couple weeks ago. When
I first put it together, the system ran 24/7 doing distributed
computer projects including CUDA, so the CPU and GPU (the GTX 580)
were running full power all the time, and I never had a problem with
that.

At this point I'm suspecting either the motherboard or the power
supply (Corsair CMPSU-850HX 850 watt Silver series) - the question is,
is there any way to further determine what the problem is without
throwing new parts at it? If I could narrow it down to the power
supply, that would be a lot easier fix than the motherboard.
Unfortunately I'd have to cannibalize another system in order to get
another power supply to test it with.

Systems specs:

MB: Asus P8Z68 DELUXE/GEN3
CPU: Intel I7-2700K
RAM: 4x4GB Corsair Vengeance low profile DDR3-1600
GPU: MSI GTX 760 or EVGA GTX 580
SSD: SanDisk Extreme Pro 480GB
HDDs: 2x1TB WD Black
PS: Corsair CMPSU-850HX
OS: Win 7 Home Premium SP1 64Bit

Thanks for any help and suggestions.

Jerry