$1000 including Windows and a monitor is much different than $1000 without those things.
Basically, though, I would start with an i5-2500K, the cheapest Z68 motherboard that supports two PCIe 2.0 x16 in an x8/x8 configuration, and an nvidia 560 or 560ti video card. That might help get you started.
If you go down to $800 then you will take a major hit somewhere and loose performance...not saying the system would not be good but if you have the 1000 I would suggest staying around that ball park for a higher quality build...
non modular PSU's are cheaper but you have to deal with all the extra cables. I am glad that I went with modular for my first build as it made it a lot easier when it came to managing the cables.
For the GPU you could drop from the HD6950 and go to the HD6870 which would not be to bad to begin with. What games are you mostly playing and what do you want to get out of them performance wise? The 6950 i listed gets you a Dues Ex game coupon with purchase of the card
With the mobo its going to be hard to find a quality one even in P67 chipset that it cheaper than $129. The one I listed is the new z68 chipset and it's at a good $149 and also a toms hardware recommended buy.
If you're willing to take jeremy1183's build and add $200, you can significantly upgrade your graphics. It is my understanding that a gtx 570 outperforms anything in the $325 range, although I am not a graphics expert. Obviously the games you plan to play make a big impact on that decision, though.
I want a single gpu to start with. I'll add a second in time. As for the games I play. I play bf2, crysis2, wow, sc2, and a few others. Will play gears 3 when it comes out. Lot's of time on my hands. I work weird hours!
Which price range? From the article you linked: "The GeForce GTX 570 offers unmatched performance for $320, competing with the Radeon HD 6970 at $350." However it puts the 6950 in a different category.