I have tested the HD 7870 XT on Crysis 3. I can play at Very High settings and FXAA. Using Fraps to measure the frame rates, I only get around 33 FPS only. Gameplay is still considered marginally smooth for me, no lagging.

And yes, Crysis 3 is really dependent on your CPU too. See the graph below. Generally, Core i3 and below is bottlenecking a GTX 680 already. More read here.

First, try to OC your processor. Phenoms are generally known to be good overclockers. You should be able to minimize some of the bottleneck by doing so.

Also, bear in mind that there is certain limit on how much FPS you can get from the HD 7870 XT. FYI, the HD 7870 LE inside the graph is the same as 7870 XT.