Thanks for all the responses.
By clicking sound, when I first heard the sound, I thought it was the hard drive failing. It sounded like the metallic clicking a drive makes when it is trying to callibrate itself. But with the drive disconnected, it wasn't the source of the problem.
And it wasn't a post beep code. No beeps at all. If it was, I would have been able to to track them down easily enough. The clicks are very short and there is a rapid, but erratic stream of them that goes on for a variable amount of time.
Even though memtest86 came back clean for one complete pass, I'm going to try taking out each of the two dimms to see if that changes anything. I'm also going to hook up a replacement power supply in case that is intermittent.
Otherwise, it seems I'm down to either the mb or the cpu. Any good suggestions for techniques or tools to test either of the two?