If you're really serious about making tap water work as a coolant there's a very easy way - but you still need a pump for your loop.
Assemble a normal WC loop. Get a 3-5 gallon bucket and insulate it, including a good piece of foam for a lid. Fill the bucket with tap water and submerge your radiator in the water. Run a hose from the tap into the bucket putting the hose outlet near the bottom and find a clean way to drain water from the top of the bucket. Turn on the tap then turn on our computer.
Another way to use tap water that's similar - trickle tap water across your rad. It won't be quite as effective as the submerged solution but if you're artistic you could make an interesting fountain around the rad and, to some extend, create evaporative cooling ...