First, I tried my old desktop PC. It was never completely quiet, playing stuff wasn't very straightforwards and it soon started losing its marbles as well.
Then I tried a PS3. Unfortunately, while powerful enough and easy to use, it was both noisy and as inflexible as I feared. It took a good amount of tinkering to get it to play some, but not all videos.
Next I tried an Acer Revo, after being enthused at by a guy at work. Turns out it was neither as quiet as I wanted, nor able to play the more demanding 1080p stuff. But it, and XBMC coupled with my Windows MediaCenter Remote were a definite step in the right direction.
Today, I ordered and collected from Scan and built a totally silent PC; there are no moving parts. It plays challenging 1080p at around 45% CPU usage in standard mode, or 10% in GPU-accelerated mode. I'm writing this entry on it. Colour me chuffed.
At present I operate it without the case lid on (but with the sides on) to ensure adequate ventilation, but I suspect I can make some chimneys and holes to give it a somewhat industrial look.
The key components that made it noiseless were the AMD Athlon II X3 EE chip which is 45W, combined with the Thermalright u120 heatsink, and the fanless (well, except I'm guessing for overload, which hasn't happened in my testing) PSU, and the Intel X-25V SSD, which is ~£80 for 40Gb of pretty fast (but not as fast as the X-25M) SSD.
The motherboard has the AMD 880G/SB850 chipset which means there's a Radeon 4250 integrated graphics which is fanless but has VGA/DVI/HDMI out.