Engine warm up makes no difference (at least on my Tahoe). The noise occurs until the differential is warmed up, and the engine does not warm up the differential. It needs to be driven to warm up the differential.
I changed the fluid in mine today. I'll update with whether it made a difference after I put some miles on it.
It took a little bit of driving but the noise has gone away with the fluid change. For reference, mine started making noise around the 45k mile mark, which is when they recommend fluid changes in the tranny and differential.