I actually thought of that -- using a mini-fridge as a case -- myself awhile back. I was walking thru an Office Max store and saw one and thought, "Hmmm ... I wonder if ...." I just didn't have $150 to plunk down at the time. 😛
Like Pathogen03 says though, you wouldn't be able to air cool with a mini-fridge (unless you cut a hole in the thing, which would then let the cool air out and defeat the efficacy of the fridge cooling). You'd also have to find just the right temp where things would be cool enough but not so cool as to form condensation. Obviously, you don't want water droplets on your components!
Only probs I could foresee with just mounting components to the underside of a desk would be:
1) Noise -- no sealed-up case to keep the noise inside
2) No grounding -- some components rely on the metal frame of the case for at least part of their grounding (I suppose you could use ground screws and some wire to ground things to something in the room, however)
3) Dust
4) Vulnerability to damage -- exposed components vulnerable to an errant knee, pets, kids or whatever, with accompanying danger of electric shock if some components were touched while on
5) A burned mounting surface -- if you mount some hot components to a wood desk, they may just generate enough heat to actually singe the wood (I suppose you could put tin foil between them though)
Some of these probs could probably be overcome, and it might be an interesting experiment to try. Let us know if you do. 😀 Personally, I think putting the computer somewhere where you do have room for it and then just using longer cables would be preferable to any of the aforementioned extreme measures. But it is fun to think about, huh?