Wanted to let you know what the problem was. Turns out my motherboard,
http://usa.asus.com/Product.aspx?P_ID=szSFtrap7crpBaQE, only supports either SLI or a single card. The second slot is not powerful enough to run a card on it's own. Go figure. Thanks to those who helped.
I have an Asus board too M4A89GTD PRO/USB3 and I cannot for the life of me get 2 5970s to work. I have purchased a new "Crossfire certified" Corsair 1200 watt psu, along with a MSI 890FXA-GD70 AM3 motherboard. I will test the new psu on my current mobo though to see what the problem is.
edit: Sorry, I forgot to mention I am having the exact problem you are: no video signal to my monitor while the fans and everything else seems to work fine. I believe these mobos just don't support crossfire that well for some reason. I did originally have a dual 5770 crossfire setup previously, so I know those work.