You did read the apple support doc? It talks about 30-inch monitor resolution, and for high resolution you need dual link active DVI.
I notice you don't mention the size and resolution of the others, and your HDMI is limited to 1080 so not an issue as long as proper adapter.
Apple Mini DisplayPort adapters: Frequently asked questions (FAQ)
Guide to Apple graphic display adapters needed
Atlona (Dual Link) DVI to Mini DisplayPort Converter for Apple 27-inch LED and 27-inch iMac
Mac Pro (Early 2009), Mac Pro (Mid 2010), Mac Pro (Mid 2012): Issues with three displays and multiple DVI, HDMI connections
It's not the USB data that is needed to run three displays. It's the extra POWER. If you read the Apple article, they say the only solution they endorse is their US$100 Dual-Link adapters. Check the table at the end of the article.
Readers here have found that if you have two Brand-X Powered adapters, aka active adapters, it also works fine. These do not always require another USB. If designed recently, they can pick up a power pin in the Mini DisplayPort interface.
AMD (makers of the 5770 and 5870) lists the qualified POWERED adapters that meet their most stringent requirements. Look down the page and you will find several qualified Active Mini DisplyPort to various others, such as HDMI. These adapters are available from other outlets, such as Amazon, sometimes for far less than the Apple adapters.
The other desktop monitor will plug directly into the DVI port on the card and the 40 inch tv will still connect via HDMI over the second Mini Display port.
That will work only if you use an active Mini DisplayPort to HDMI adapter.
The only (someone) have found to work (2009)
but you should confirm that they are really active before buying 500 of them.
You can connect a DVI to HDMI adapter or cable to an active Mini DisplayPort to DVI adapter but that won't give you any audio over HDMI. You could connect the DVI displays to the Mini DisplayPorts with active adapters and use a DVI to HDMI cable or adapter from the DVI port to the TV.
If you want audio via HDMI, you can use audio from the Mac Pro's optical output with this DVI to HDMI converter:
You only REQUIRE Dual-Link DVI adapters when your displays are wider than 1920 pixels wide.
What is in short supply is Power to run signal Boosters from converted Mini DisplayPort to DVI (or its re-wire, HDMI).
What you NEED is two ACTIVE Mini DisplayPort adapters. Apple's only ACTIVE adapter is their excellent Dual-Link DVCI at the amazing US$100 price-point.
But ACTIVE Mini DisplayPort to Single-Link DVI adapters are available for US$35 to $40 from sellers such as Amazon and NewEgg from Accell and Startech:
Adapts: Mini DisplayPort™ To: DVI-D Single Link (Active Adapter) Manufacturer Part Number Firmware Max. Resolution Purchase Accell B087B-006B 0.95 1920x1200 Buy Now Astron 3411020-090-H 0.95 1920x1200 Buy Now Bizlink KS30015-131 0.95 1920x1200 Startech MDP2DVIS 0.95 1920x1200 Buy Now Wieson G9858C219-006-H 0.95 1920x1200 Buy Now
There are now active Mini DisplayPort to HDMI adapters
Apple is suggesting the dual-link DVI adapters because they don't sell active single-link DVI or HDMI adapters. If your display resolution is not over 18920 x 1200, dual-link is not needed if you get non-Apple active adapters, and you will save money and not tie up extra USB ports.