Have you solved it yet? because I'm experiencing this as well and so are other 7870 users. Just type in 7870 black screen and there will be many results.
NO it's not your PSU... I'm using FSP 600W Aurum 80+ GOLD and I'm experiencing this too. and yes my BIOS is updated and all of my drivers (sound,GPU,Game). tried it from 12.3 to 12.7 beta and it still crash.
all parts of my PC are NEW. using ORIG game and OS.
I have returned my card to the shop I bought it in and it was tested by the distibutor and they can't find anything wrong with the card.
It passed GPU benchmarks, memory test, prime95...
it was just returned to me w/o any fix.
I'm still having problems with the the 7870.
I haven't experienced crash when only using the integrated graphics.
obviously the problem is with the 7870.