Help! Horrible noise while accelerating

I have a 2003 PT Cruiser GT with Automatic and 60,000 miles. Under moderate to hard acceleration I hear a grinding type noise. I have taken it to two shops and they are unable to figure out what it is. The mounts appear fine as are the wheel bearings. One of the shops put a scope up to the turbo and said it is quiet. It makes no noise when revved in park. Once up to speed the car does not make the noise very much at all either. I noticed that it also seems to decrease as the car warms up. I can actually kinda feel it when it makes the noise as well… The transmission shifts smoothly through all of the gears so I do not think it is that. This is very puzzling and I just need to have it fixed. Thanks in advance.

Has anyone inspected the exhaust system for loose heat shields? This is a very common thing.

I agree that this is very possibly just a case of a loose or damaged heat shield.