I have a 1999 Honda Civic. The noise began quite a while (couple of years) ago. The noise is sporadic in that it does not happen all the time. But when it does happen, it seems to be when I first drive the car after it has been sitting for a while—i.e.- cold car. As the car warms up, the noise goes away. Originally noise sounded like a piece of metal dragging on the ground and the noise seemed to be coming from passenger side wheel area. I had it checked a couple of times—looked at brakes, padding, etc…and for anything that might be loose, but no one could find anything.
Now the noise has grown from the sound of single piece of metal being dragged on the ground to a much louder, larger piece of metal (or several pieces of metal) dragging on the ground. No noise when car is idling, neutral, first or any gear. But noise will appear with moving forward, backward, turning. Once car warms up, noise stops. Noise still seems like it is coming from front of car, but no longer just the right wheel area. When the noise is there, there are no warning lights on the dash. No noise, when I rev engine in neutral. Just when the car moves until the engine warms up.
While this is very difficult to diagnose from a distance, my first suspect would be a loose heat shield or two. While not dangerous, the noise from these pieces of sheet metal can be extremely annoying when the engine is cold, and they do frequently tend to go away when everything is warmed up.
Have your mechanic check the heat shields the next time that the car is on the lift for an oil change.
Note: Do not ask the kiddies at Jiffy Lube to check this, as they can barely find the oil drain plug on many cars.