I have noise coming from my fan belt on my 1999 Corvette. It doesn't appear to be loose. It's only visible when first starting the car and goes away once the car is warmed up.

could be a worn belt , or alternator pulley, waterpump pulley, dirt road grime, etc., check your BELTS to make sure they are ok, not chewed up, worn, loose , replace if necessary, keep an eye on it. don't worry about it,

Make sure all your belts are tight and in good shape. If the inside surface on the belt is shiny and not dull it has been running loose and is "polished". If this is the case replace the belt. If that is not an option for you try some belt dressing or the old standby, use bar soap and coat the inside of the belt. If this doesn't help you may have a bad pulley on the compressor itself. If the air works you either live with the noise or replace the pulley,

It can be also due to poor belt or tightness. When the engine is cold pour some water over the belt , if the noise goes off then the belt can be a suspect. Also you can use a candle to rub wax on the edge of the belt and then start the engine, if there is no noise then the belt can be a suspect.
If not check on the IDLER pulley or the bearings in the water pump/alternator.
check this link:

http://www.fixya.com/cars/r5816171-squeaking_belt_noise

the only belt that stopped my chirp was a gator belt

belt see if loose

try spraying some preferably graphite on belt or wd40

Your belt may be worn & slipping. Check on the inside of belt for lost belt v grooves. If O.K., you may have a a bearing starting to sieze i.e. on alternator, power stearing, air conditioner condenser, or idler pulley. You will be able to check this by slackening belt & spinning each pulley. Good luck & Merry Christmas!

Cooling fan relay would be a good starting place.

there is a weap hole in the bottom of the pump if it is leaking from that it is the pump they will make noise when they go out also

try checking the tension on all the belts(fan/water pump, power steering, alternator). If loose, they'll whine.