My Audi A3 with automatic transmission is emmiting this weird sound when I let off the gas at slow speeds. I think the car shifts up to a more efficient gear (probably 3) thinking I don't need to accelerate quickly, but it is accompanied by a thud type sound in the background. I wonder what I can do about this and what might be the possible cause before more harm is done.
I'm thinking the issue is probably with the torque rod (on some cars called a dog bone) which keeps the engine/transmission in check during acceleration/deceleration. At least one end of the "dog bone" has a hard rubber piece which can wear out over time. Depending on how worn out it is, the noise will could be louder or not. You would probably notice it will start getting louder over time as more and more of the rubber starts to tear.
Here is a picture of an aftermarket (billet) piece. It is for an 1996 - 2003 Audi A3/S3 (8L). Yours should look something like it (depending on the year), only not nearly as pretty :D
Here is an image with the location of the dogbone. It is #8 on the diagram and appears as it is right up front on the A3: