2011 RX350 152,000 kms (94,400 miles)
Got it a month ago, and it ran fine until yesterday when it started vibrating when accelerating at any speed.
Becomes smooth again when I back off the gas pedal.
From what I can find online this is a known issue
Short of taking it to a dealer, what steps are recommended to try to narrow down the cause?

 

Take it to an independent Toyota specialist instead of a dealer who will overcharge and sell you unneeded repairs. The RX is a mechanical twin of the Toyota Highlander, Venza and Sienna.
Your vibrations could be many things such as bad gas, a bad motor mount, transmission mount, ball joints, wheel bearing, rough engine due to some unknown cause or a transmission issue. If AWD it could be other things involving the driveshaft or transfer case. Find a good local independent shop that specializes in Toyota or Japanese vehicles.

One test would be to put it in neutral and hit the gas while rolling to see if the vibration comes and goes. That may help isolate the motor or transmission mounts.

 

I have had past experience with this on an old Toyota Harrier V6 which is similar to the early Lexux RX330.

It was worn driveshafts that had that symptom. So this might be the same reason for your situation.