A couple of questions and ideas:
You said it doesn't depend on speed, but I assume that the noise stops when you stop the car, right? So that would mean that it something related to the chassis motion caused by driving, and not related to engine rpm or drivetrain rpm. First eliminate the relationship with engine RPM by letting it coast in neutral with the engine turned off. If the noise is still there, you've ruled out anything related to the engine running.
How about getting the car warmed up to the point where it's rattling and then getting someone else to drive it while you climb around the interior, open each window and listen for it from different areas? This may help you isolate whether it's really coming from the dash versus the underside, engine compartment, etc.
Is this a repeatable problem, or intermittent? Can you replicate it anytime? If so, take the mechanic for a ride when you can demonstrate the noise.
Hope this helps...
Experience is something you don't get until just after you need it.