I had the exact same problem. I had the Sapphire 7870 OC and at any random point, my screen would go black and not respond to anything with an annoying sound loop. Turns out it was poor-quality capacitors that AMD used for testing. They realized they sucked, but didn't get the word out in time, so several manufacturers have this problem. It's impossible to fix, but on the good side a lot of manufacturers have been very lenient on the warranties and returns since the discovery of the problem.
I was lucky and managed to get it back to Newegg within just a couple of weeks. Now I have a gtx 670 ftw and couldn't be happier So basically, you'll have to RMA the card. It's a hardware problem that can't be fixed