I doubt there's a low-cost solution to this. I know that DisplayPort carries audio and if you got an Eyefinity 6 card, I think it is possible to hook 6 monitors with DisplayPort and audio support into the card, each with their own independent audio stream.
Not sure if there is a max cable length for DisplayPort though.
I'm sure there's a better solution than using Commercial graphics cards though. Although it would be more expensive.
If this won't be used for official medical reviews (those must be certified monitors), then you can do with any monitor you like. Even a TV would be a good idea
There are hubs to support multi monitors with just 1 input, so you can look around for one. If you need/want them to be independent, then you either need a computer with mutliple display support (AMD Video cards usually give you 5 to 6 monitor support on their HD7000 series and FirePro cards) or a little computer for each (like an HTPC; zotac has good ones for this).
Going with a professional Video card, would skyrocket the price, but will ensure you to have 6 monitor support if you pick the right card (either a Quadro or FirePro will do fine). For a consumer level card, there are Radeon HD5870's with 6 monitor support and other special editions, but don't remember any right now.
There are distance limitations on video cables that you are going to definitely exceed running to 6 different rooms. You also are going to need some kind of special software to manage 6 different videos and 6 different displays considering you can't see the monitors.
You know, for me, the simplest solution might be to install TV's with build in DVD or BluRay players and burn up a disk that has the clips on a repeating loop or have the medical assistant prepare the TV with the right specific one while they are cleaning the room from the last exam.
You haven't mentioned any distances so I'm not sure what limitations you would "definitely exceed"...
You can run VGA through high quality shielded VGA cable for at least 20m (tested it myself).
If you want to go further, you can get cheap VGA amps, usually a VGA splitter/amp will do the trick, allowing you to send the same signal to 2, 4, 6 or more monitors. Make sure the cable and connectors are good quality and shielded to avoid signal degradation. The audio will run easily for 20m, but shielded and balanced signals are even better. Use a balun if you find there is audio noise.
However, running a different feed to each room will be a problem unless you look into hardware and software for the specific purpose. Otherwise you're going to have an impossible time sending 6 different a/v signals to the correct screen with a media player that was built for single screen viewing.
Hardware wise, you would need as many independent video and audio outputs as you want - usually they come 2 per video card, sometimes 3. A motherboard with multiple PCIe slots and matching video cards should do the trick. You'll also have to sort out matching audio outputs.