| I've been doing something like that from MythTV for a few years now. Yes, it's Linux, and yes, it does offer TiVo functionality.
If you're just after a machine to just serve media content, it doesn't generally need a lot of processing power, just bit disks. That's usually required for the machine that's doing the decoding. There's a fair bit of software out there.
Don't forget that you will need a fairly good network to stream across, and that any TV is going to need something to interface to it.
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