I haven't played D3 in a while but I seem to remember maxing it just fine as long as AA was off. When enabling AA I started to see framerates deteriorate at 1920x1200.
A 7850 is a faster card than a 5870, OP's card is probably sufficient for what he's asking but only just, and dropping a slider or two might help keep things fluid when there's a lot going on. I would like to suggest that he could benefit from an upgrade, but anything more than a 7950 would be silly if D3 is his only goal.
A CPU upgrade will almost certainly help.