Your experience is very close to mine. The clicking sound started about a year ago, first just 2 or 3 clicks, then taking longer. This week it won't turn on at all. I even tried warming it up with a hair dryer, nothing works. In my opinion, get it fixed while it's under warranty. The problem won't do anything but get worse. I'm going to try to see if I can get a part and fix it myself. I hope this helps.
I've been noticing that my Samsung LNT4061F is behaving strange lately.
When I power up, it 'clicks' three or four times before actually turning on. Almost like someone is turning it on and then off and back on a couple times.
At first, I thought, perhaps it's trying to synch, but I don't think that is it. I don't recall this clicking happening like this before.
I bought this in '08 from best by and do have a 4 year plan, so I'm covered I would think. However, because it DOES turn on at this point, I want to make sure this IS a problem and get Best Buy to fix it. My fear is that this will eventually turn into the TV not powering on at all when my TV is out of warranty.
I should also note. Once on, if I turn it off and then power back, it doesn't seem to do this. So, only when TV appears to have been off for some time.