1) As the budget of your 24'' monitor (1920x1080) depends on the cost of the card, I would swift more fund to the monitor as it will stay much longer with you than any graphic card will. And, 6870 is the cheapest of the 3. This is the paramount reason.
2) HD6950 has 2GB Vram and even though your OS is 64-bit which allows you to fully utilize the 4GB system ram, nevertheless 2GB Vram (vs 1GB of 6870 and 1.3GB of 570) might map too much into system memory for address and therefore gives you less system ram to use. This might not be a problematic, depending on how you use your system ram but you should keep this in mind.
3) Your mobo cannot do SLI. Therefore, when the time comes for more graphic power, you need/can go Crossfire. So, AMD has the advantage here.
A single 6870 should give more than 50 fps at 1920x1080 except in the few games like Crysis and Metro 2033. So, you have to accept this fact.
Edited by windfire - 12/22/10 at 3:30am