I drove the new fj and i loved it! Lots of power, cool interior, and it would great for off roading. The blue with a white roof is the way to go.

Noise
 

On a lot of diesel engined vehicles in Europe a secondary heater running off the fuel tank is fitted. This sounds like a jet engine shutting down when you switch off. It only activates when its cold. The diesel X3&5 has one, not sure if it's fitted to cold country petrol engined X3/5's.

Off topic I know, but...

Before getting the X3 I test drove the Xterra and hated it. Felt like it would be super easy to tip the thing over. Might have been because I just finished driving the Pathfinder. I was very happily surprised with the Pathfinder's handling. The next best handling SUV to the X3 that I have tried.

Getting back on topic, I recently spent a lot of time under the X3. I also checked the parts list and see no fans at the back of the car. The only fan that might be on after the car is shut off is the fan for the radiator. If the engine is warm enough that fan will keep running when the car is off. Next time you hear it, try opening the hood and see if that fan is running.

Good luck!

Could be the leak diagnostic pump
 

Someone answered on the RoadFly board.
I quote

"Although new to the X3, I am familiar with BMWs. There is a pump that tests your fuel evap system for leaks, and I think it activates once you shut the vehicle down. Check out part #2 at the link below. The pump MAY be what you are hearing."

Leak Diagnostic Pump

That's very plausible. I hear the same thing on my X5.