should work if cpu could handle it
An Athlon II X2 250 will handle just about any single graphics card you can buy without bottlenecking so any Socket AM3 CPU will handle a 6770.
The PCI Express x16 interface will not start bottlenecking until you move up to the highest end graphics cards. PCI Express v1 x16 runs at the same bandwidth as PCIE v2 x8. As you can see in the benchmarks there is very little loss in most situations running a GTX 480 at PCIE v2 x8.http://www.tomshardware.com/reviews/pcie-geforce-gtx-480-x16-x8-x4,2696.html
The HD6770 will run fine as long as you have a decent 400-450w PSU and will play most games at 1440 x 900 or 1280 x 1024, you might even strech it to 1680 x 1050 but you will have to start turning down the eye candy down from there on up.
In short it will work fine.