Thought I would add this thread. The fix seems obvious but it isn't always so. This morning my 04 F150 Supercrew 5.4 did the "chirp chirp" from under the hood upon cold startup. It had chirped before so I thought it would go away once warmed up this time as well but it got worse. Lifted the hood and listened for where the problem was. Idler pulley on the top right of the motor definitely grinding (or so I thought). Bought a new pulley AND belt from Autozone and installed the pulley first. Fired up and the squealing did not change. Strike one. Changed the belt, bumped the motor over about 4 times to make sure the belt was going to stay seated, then fired it up. Problem solved. Lesson learned? A person can use the screwdriver method for correct diagnosis seen in this video. Did not hurt to replace the idler either since the belt was off.
https://www.youtube.com/watch?v=QyZWhz7uvDE