Unfortunate that none have MDP or DP ports. Okay, you've got two options:
1) Use the existing 5770. Hook one monitor up to the DVI port using a cable. Hook second one up with inexpensive
passive MDP to DVI adapter. Hook third up using
active MDP to DVI adapter.
At Monoprice.com, passive adapters are about $7 and active adapters are about $70.
So you'd need to spend about $80 in adapters and you will tie up a USB port, but you'll keep all your extra PCIe slots free for other use.
2) Get a second, inexpensive card. This could cost you less than the adapters, for example a GT120 is about $30-50 used on Ebay. Hook two monitors up to the 5770 and the third up to the GT120. However some people advise not to install AMD and Nvidia in the same computer, so you might look for an inexpensive AMD/ATI card instead. You'll tie up another PCIe slot, use more power, and generate more heat. If you get a card with a fan you will probably add noise as well.
Also note that if your monitor exceeds 1920x1200 resolution, all of your DVI cables and DVI adapters must be dual-link DVI, not single-link DVI. As far as I know there are no passive dual-link DVI adapters.
Personally I'd go with option 1 but both are valid.