I doubt any Apple graphic card came with HDMI.
Adapters yes. MDP or DVI to HDMI
The OS you use helps too. And Mac Pro model (3,1 etc)
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: