However, with the old 560 in there, I've had four random shutdowns. All have happened while I've had the card "doing stuff"; two shutdowns when running Dolphin and two in Warcraft. The system just spontaneously powers off, then about five seconds later it boots up again.
I've reseated the card and while it's warm to the touch, it can't really be described as "hot". I've confirmed that all of the computer's fans are running, including the video card fan. The only thing that's changed since it was working is the EFI ROM version; I had to upgrade from "F8" to "F13" to get the system to recognise the 7950. The board is a Gigabyte GA-Z77M-D3H.
So far I haven't had any shutdowns when not running games, so I'd guess that it's either a heat or power issue, but the hardware has been completely reliable since I bought it last year.