Sound propagates at various speeds depending on what it's traveling through. If you can get an aircraft to travel faster than the speed of sound through the structure I suspect it would be silent. In the case of good old fashoned aluminum (which is a hair over 5,000 m/s or more than 10,000 miles an hour) you'd have to move pretty quick.
As for whether or not the movement of the structure matters, I'd say it does; you're still propagating a pressure wave, and if the medium the wave is moving through is ALSO moving it will not propagate at the same rate. Now, of course, I have to start digging for references

I've only been in a couple high performance jets; they were plenty noisy on the other side of the brain bucket. Our resident F-111 pilot assures me this is normal.