Have just been through similar thing with mine and ended up being the tyres. Wasn't dependent on turning though - a weird rumbling/resonating noise between 55 and 65mph (exhaust too noisy after that). Never felt any vibration and the noise seemed to be coming from under the car somewhere.
Took it to Audi and they said it was down to 2 of my alloys not being round anymore. Wasn't dangerous so I wasn't going to shell out a grand on 2 new wheels.
Got a tyre replaced on Monday just gone (all 3 others replaced in last 3 months) as the inside edge was badly worn and the noise is gone.
Apparently the TT is bad for this and GoodYear F1s (which mine has on) are paricularly bad and deform after a while.