Chuck – So you mean there is an issue on the supply side between the house and the main in the street?
I was just pointing out that water coming in from the main will be at ground temperature, perhaps 10-15°C cooler than it is inside your house. If this is the mechanism there would probably be a slight delay before the clicking started and it would gradually slow down after a minute or two of running water.
Another possibility is your water meter is knackered. Here's a recording of a fairly noisy one:
If this is the issue the clicking rate will vary with flow in a fairly repeatable and persistent manner.
Noise can propagate quite quite well along a pipe full of water.
There are doubtless other possibilities but those would be high on my list of possibilities to consider.