For a combined box I would say the following:
2.5+Ghz processor. Faster if want to encode more than one stream at a time. Dual core is also a huge plus.
nVidia graphics with TV out -- note you can find this integrated into many motherboards, even cheap ones.
sound -- the integrated audio on the motherboard will work. Some of CL's latest Audigy cards will not.
optionally supported tv tuner, I'd recommend haupauge and/or an HDHomeRun
optionally voip wifi phone
optionally unlocked linksys pap2 or similar
optionally lots of storage in a RAID configuration.
optionally a gyro mouse, wii mote w/ bluetooth dongle, or supported remote/receiver.
optionally a supported IR blaster to let your unit control your tv
optionally a bunch of x10 light controllers, video cameras, etc.
Some notes -- any raid controller under $100 will not do hardware raid. This includes raid controllers on the motherboard. These are "fake raid" controllers or "windows raid" controllers. The let you configure the drives before the os loads, but the driver still does the RAID in software. Linux treats these like plain IDE/SATA controllers and forces you to do normal software RAID, so you can ignore "raid level" as a feature unless you buy something like a 3ware true-hardware raid controller
I highly recommend nVidia graphics. There are a lot fewer headaches getting it working with 3D acceleration and TV-out than ATI cards. Intel's newer chips support decent 3d acceleration and come with OSS drivers, so they should be a snap as well, but I'm not sure how to separate the good from the bad.
Many TV tuners support hardware accelerated compression. Most Haupauge cards will do mpeg2 (dvd) compression, freeing your CPU from that work. I prefer mpeg4 (divx/xvid) for the smaller space, thus my cpu still does all the work. There is at least 1 tuner that does mpeg4 compression in hardware; I believe it's a USB device.
A lot of your requirements really depend on what you're trying to do exactly. Also, mythtv runs on ubuntu. There are is a cohesive set of setup tutorials on wiki.ubuntu.com.