I can't help feeling that this could be a great opportunity for you to build yourself an ITX design Media Center.
It may be a little too 'enthusiast' than what you had in mind but regardless, I have one in my living room, and it is fantastic!
They are basically, TINY motherboards with inbuilt CPUs, mostly made by VIA.
There are different designs and connections to suit absolutely any application (including dual monitor). Most of the board are completely fanless for a silent system, and there are plenty with graphic acceleration, optical sound output and even HDMI (even though you have said this is unnecessary)
I've had a quick look for you, and I believe the EPIA EX 10000EG Fanless C7 to be the best option for your particular setup. It has everything you have requested on your list minus a TV Card.
At £129 it may seem a bit steep, but don't forget you are getting the board and the processor and the graphics for this price.
There are a huge variety of cases on the site, although some of them tend to get on the pricey side. If you fancied turning this into a bit of a project, you could make your own case out of... well just about anything.People have used old PSOnes, guitars, DVD player cases. Have a look at http://www.mini-itx.com/projects.asp for inspiration.
You can also pick up all the required drives (DVD and Hard drives) As well as any adaptors required.
All in all, from a quick spec I ran up, you could have the entire system minus the TV card for between £220 and £250 depending of spec and drives etc.
And I know you said you wanted to use normal drives, but I would seriously consider using a laptop (2.5") drive for a media center. Speed won't be an issue, and you don't want to hear the constant clicking and vibration from a larger, faster drive.
Also, it may prove useful to take a look here for advice on connecting your TV card to different boxes and equipment. There are also some recommendations for different TV capture cards: http://www.radioandtelly.co.uk/onyourpc.html
Just make sure your TV card fits your case! Maybe a USB is the best bet.
Anyway, this is long enough now. If you'd still like to do an AMD setup, I can give you some advice on the board and CPU but it will obviously be a much bigger setup and will almost certainly need a fan for the CPU as a media center case will be too small for a passive heatsink cooler. Also you will have to use a standard PSU which will also have a fan.