I've done it a couple of times, for the same reason you want to do it.
The mounting posts in the new case must line up exactly with the holes in the big circuit board you want to transplant. They usually do with standard ATX PSUs, except Deltas. Don't bother trying the transplant otherwise.
There has to be sufficient clearance for all the components, and usually any problem is with the AC power cord receptacle at the back, especially if it's low in the case and sticks in far because of its line filter (either a tin can or small circuit board soldered to the receptacle's tabs).
It's probably best to use the on/off switch of the new case because removing it often causes at least one of its plastic retention tabs to break off.
You'll probably have to solder the wires for the on/off switch and AC receptacle but maybe also for the fan if there's no socket for it.
The original PSU may have had sheets of plastic electrical insulation under the big circuit board and maybe also on the cover. You may want to transplant these, too, or use fish paper (not for fish but electronics; much like paper for automobile gaskets) or the plastic from a 2L soda bottle (generic is better -- perfectly cylindrical).
Some of the big heatsinks are connected directly to 340 volts DC when operating, so make sure everything clears them by at least 1/4".
Do NOT operate the PSU except with the PSU case completely secured to its cover, with all the screws in place. Use only a grounded 3-wire AC socket.
Edited by larymoencurly - 6/17/13 at 4:48pm