2006 Toyota Corolla Early Morning noise

When I first start my car in the morning and put it in reverse and then drive I get a gringing, vibrating noise. When I put it in PARK it stops. It sounds if it is coming from the rear. After I drive a few miles the sound stops. I took the car to the Toyota dealership and they could not find anything wrong. They hooked it up to their computer and the tech and the service manager did a test drive. I told them this wasnt't normal. I took it to a tire store and they put on 4 new tires. The sound is getting louder. What would make this noise in the morning and obviously when the car warms up the noise stops?