I've Atlon II X3 440 and 4GB DDR3 RAM. The 'budget' motherboard should have a decent onboard ATI Graphics which will run Games like Call Of Duty MO at medium to high settings. Brands- Biostar, Asus, Asrock, GA.
I think youre going to struggle running any game at a playable level with onboard graphics, if youre on a major budget, maybe try buying a cheap mobo and a 2nd hand GPU
Any MB you will get, will have a very poor graphics chip on it. you might game on extreme LOW settings, but thats about it. Your CPU isn't that beefy either as is a Rana core, so it has no L3 Cache which doesn't help in gaming.
the AM3 motherboards with on board GPU's have either a ATI Radeon 3000, HD4250, HD4290. none of these are suitable for gaming.