Wyatt, it's almost never worth updating a rig like that, it's usually a pain for lackluster results. Just keep your money and save it for a custom rig later on in life. Besides, the 5670 is "ok" but the 5770 is much better and doesn't cost too much more(Of course, the 5770 isn't too fantastic for playing games, which is what I imagine you'll want a new GPU for). 5770 costs like 120 or something nowadays. If you do go along with it, remember to check your PSU and the power draw a 5670 has and see if you would need a new one.
Besides, if you're going to go AMD, it's pretty much go big or go home.
http://www.youtube.com/watch?v=usGkq7tAhfc