When my wife starts her truck there is a chirping sound that comes from beneath the engine compartment. It sounds like a bird chirping continuously. It does not get louder with acceleration, it just stays at the same level. I honestly can't say that it happens during acceleration because we can only hear it while it's idling. I've had the serpentine belt along with a tension pulley changed but it still persists. The last tune-up was performed at about 110,000 miles.
Just taking a guess here, it might be your water pump. Are you losing any coolant? That would be a sign too. With the engine on, stand to the side of the engine compartment near the front of the engine. Stand where you are exactly perpendicular to the serpentine belt and pulleys. Look very closely at the belt and look closely at the pulleys one at a time. Do you see any wobble on the water pump pulley? Or any of the other pulleys? There shouldn't be any wobbling or belt deflection. With the car off, take the serpentine belt off and check the pulleys one at a time. Grasp a pulley and try to go up and down with it. There should be any play in it. Spin the pulleys by hand and see if any of them make noise.