I like the 6870 the most personally, but it's fairly on par with the 560, so buy the cheaper one if there's a price difference. Although the 480 is a great performer, I never would consider getting it due to its heat and power consumption.
Well, I would like to explain my choice of the 6870, given that I use ESXi on my main machine and require pci passthrough which with gpu's only work with ATI cards at the moment
The 560 is an OCed 460 so it really doesn't compare to the 480. And the 6870 competes with the 560... so unless there are power supply considerations the 480 is obviously the best performance of these options.