I wanted to share my success in case it helps someone else. This thread was very helpful in getting me as far as I did (thanks to Grant Bennet-Alder) but my set up was not represented. I just today set up my stock Mac Pro (mid-2012) with Radeon 5770 using three LG 25UM56-p 25" ultrawide monitors (2560 x 1080) (link to NewEgg) (that have HDMI ports only). I wanted to avoid the mini DisplayPort (mDP)–to–DVI-D adapters so I looked for a straight mDP-to-HDMI solution.
1 x DVI-D
From the computer, the DVI-D port uses a DVI-D cable connected to a DVI-D–to–HDMI adapter. I don't remember where they were sourced but a least one came with another of my Apple computers.
2 x mDP
Each of the two mini DisplayPorts have Cable Matters Active mDP-to-HDMI adapters (link to Amazon) connected to HDMI cables. For any monitors over 1920 x 1080, they not only need to have active adapters in all three connections, but [I think] the mDP adapters must also be capable of higher res (these adapters are 4K capable).
But wait ... There's More!
The video card does not have firmware that works with higher res monitors so to get the image to be legible you need SwitchResX. Not the easiest software to figure out (I still don't know what finally worked setting up one, then moving to two, and now three monitors, but every time was a bit of trial and error). My settings in SwitchResX have the two mDP port monitors set to 60Hz but the DVI-D connected monitor is set to 53Hz (it might actually be 52.52Hz). If I try to make it 60Hz, too, the image is horrible.
And everything seems just fine when I boot from my Windows 7 HD on this Mac.
I hope someone finds this useful.