This is entirely and absolutely possible.
people do this with stationary motors all the time. Ever since they had motors. If you have a diesel or gasoline motor that you use for running pnuematic pumps or generating electricity or something like that then having a water pump is actually a _bad_ thing.
The reason you need water pumps in cars and such things is because everything needs to fit into a engine compartment. Having a water cooling system without a pump means less things to go wrong.. it's inexpensive, effective, and have much higher reliability. No pump to break, less lines, less places for water to leak.
The trick is something called the thermo-syphon effect. Hot water rises, cool water sinks.
So you have it going both ways.. cool water wants to go down, hot water has to go up. So you have to carefully design your system to take advantage of this.
So your not fighting gravity or anything like that. It will work just like any other syphon, the weight of the water is what drives it.
So you have the radiator above the heatsource. The 'hot water' empties out into the top of the radiator. The cool water empties out of the bottom of the radiator. (so obviously you can't have a rad were the input and the output are the same height.)
It's tricky to design it so it works correctly. For example if you have a even very small dip in a mostly vertical stretch of hose then hot water will get trapped there, much like a air bubble would, and it would destroy the syphon effect.
But once you have it designed correctly, all the hoses and the radiator correctly sized, and have it tested then it's going to be dead silent.
In fact the more heat your dealing with.. the greater the effect. If its correctly designed and the mass of the water never allows the thermo-syphon to work that just means everything is running to cool to realy require a pump in the first place.
Obviously your probably figuring out that whatever your going to build is going to be fairly large. And it definately won't be a tiny thing.
But here is a example of somebody pulling it off:
He uses his water cooling system on the cpu and the powersupply. He uses a oversized heatsink for the vga card. He uses a store-bought waterblock for the cpu.
Unfortunately he ended up needing a slow fan for the power regulators for the cpu, but I expect that slices of a old aluminum heatsink or something like that glued to them would be enough to fix that.