What GPU can run all games at 1080@60fps that costs $350 or less?

[Exzodian]

FX 6300 or FX 8350
60Hz or 120Hz Monitor?
Can a 60Hz play show above 60fps?
Crysis 3 at 1080@60fps maybe all ultra settings? Nothing above 1080 like 1440 or 4k
Will be playing Skyrim with mods and B4 also most modern games like Dying Light
If no card can hand Crysis 3 at 1080@60fps under $350 or $300
2015
Was planning on a GTX 750 ti ftw then new GTX 960 came out (ASUS or EVGA) maybe Ti in Summer?

 

[Formagella]

Looks good, I changed the motherboard out to a H97 board as you can't overclock the 4590 anyway.

http://pcpartpicker.com/p/4LQDTW

i5 4690k instead

you have to decide first whether you want to overclock or not as that decides whether you get the CPU with the k and which motherboard chipset you need. When in doubt, I'd spare the money because the GPU is going to be your gaming fps bottleneck anyway, there's not much to gain by overclocking the CPU. Also don't buy another cooler for the CPU, as high end graphic card coolers make more noise anyway. You can always buy one later if you feel a need for it.
RAM and motherboards are two areas where you can spare money if you don't need the features, for RAM the performance difference is like 2-3% in limited real world use cases going from default speeds with XMP disabled up to high end memory sticks.
It's better to spare the money and spend more on the GPU or save it for a GPU upgrade 2-3 years from now (with that PSU there's no worry about upgrading).

The last derza configuration (http://pcpartpicker.com/p/4LQDTW) looks good for all these reasons imho, except for the keyboard.
Slim keyboards really aren't as nice for writing and gaming as normal desktop keyboards or even mechanical keyboards. It's like laptop keyboards, chiclet keys. There's not enough feedback for comfortable fast typing and for gaming, and if you play intensively the most stressed keys are going to break off at some point. Get a normal one. I have a quite old keyboard (http://www.amazon.com/Compaq-Kb-0133-Windows-Keyboard-265987-008/dp/B0007VBWQ8) and it's even better than the modern standard 20$ microsoft keyboard, the feel is better imho.

With a 60 Hz monitor you won't get more than 60 images per second on it, but modern games with the highest settings will run below 60 fps anyway so you don't miss out.