The 1200 will fit the 12.16" length of the 5970with one caveat. The 1200 provides two optional fan mounts, which are ideally placed for twin GPU cards. One is on the case window and the other is in fan cage on the back of one of the HD cages. There are three HD cages on the 1200. If you mount the fan cage on the back of the middle HD cage, I'm gonna bet that the 5970 hits the cage.
If you mount the fan cage in the top HD cage, then you gonna make your 5970 very happy. The top HD cage will now have two fans in series bringing in fresh cool air over the top of the 5970.
You could also mount the fan cage on the back of the bottom HD cage in which case it will blow copious amounts of air under the card, providing plenty of fresh cool air for the card's own fan intake.
The HAF 932 is also a great case.... it and the 1200 share pluses and minuses so it comes down to what pluses ya like, what minuses don't matter to you and aesthetics. The 932 has some nice assembly features that I like (less screws) and has 3 pin fan connectors. The 1200 has molex connector on the fans and has air filters. Though I do like air filters, what did it for me was the ability to add the optional fans for the twin GPU card and the ability to take CPX form factor PSU's.
Assuming ya need a PSU along with the case, here's ya options allowing for futire XFire and overclosking: