Screeching/Whine from engine

I recently bought a used 2006 Honda Element.  Good price, interior chewed
all to hell by a wolverine or whatever the previous owner had, but nothing
seat covers can't fix.  160K miles. Manual trans.  When I start the car there
is a whine that seems to come from the right side of the engine.  High pitch,
almost like a screeching.  When starting the car cold the whine is there
regardless of what gear it is in, but it is most prominent in first and second.  
Accelerating doesn't make the noise louder but the rate of the noise
increases with acceleration.  After a while of driving the noise will disappear,
sometimes showing back up at slower speeds.  But disappearing entirely
when in neutral.  Depressing the clutch will also get rid of it, but only after is
has been driven for a while.  In the morning when I start it, the noise in there
regardless of gear or hitting the clutch.  I've read it could be a belt or a
bearing.  I can't really replicate the noise for the dealer, because by the
time I've driven it out to them, it isn't as loud or isn't making the noise at idle
like it does cold.  Any thoughts?