7300 is over 7 yrs old has 256MB and lacks some of the newer feature specifications.
Some more info about using 3 displays and when you do and do not need active or dual-link (and your 23" ACD doesn't)
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.
http://support.amd.com/us/eyefinity/Pages/eyefinity-dongles.aspx
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)
http://www.alibaba.com/product-gs/337561577/miniactive_display_port_tohdmi.html
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: