This happened to me too on Wednesday. Same 2008 Mac Pro, same card, same freeze, eaither before or after, most of the time I get the grey screen of death, "You need to restart your computer".
After a bit of searching, I reckon the video card is cooked. So I'm currently on the hunt for an ATI Radeon HD 4870. I can always go for an ATI Radeon HD 5770 or 5870, but then I'd have to switch to 10.6 (from 10.5) which I'd rather not do.
Post back if you get a fix.
Before the card was permanently cooked, I had issues, which I thought were overheating, so I switched off for a while, then removed the side panel, cleared out the dust bunnies and I got it running with video again for another 12 hours overnight, ironically long enough for me to load another 300GB of files for processing. If you get this chance, maybe use it to get OS X 10.6 on in case you need a brand new video card.
I'm able to still use my Mac Pro "headless" through screen sharing.
Hoping someone else has a similar story with a different outcome involving zip-lock bags and freezers.
If you want to have some fun with the card before recycling it, try this in a well-ventilated room:
Hold the barbecue sauce.
Old fashioned components in plated-through holes were bathed in liquid solder pumped up from below the board to solder them in place. That solder was about 450 degrees F. We have been heating electronic components (powered off) to high temeratures for short durations for a long time. It does not kill them.
Modern surface-mount components are fixed in place using solder with a much lower melting point. This process ever so slightly re-flows the solder and heals any cracks that may have developed over time.