Similarly maybe, a Norwegian tune in 3 I play once in a while, has a strong beat on 3, as I'm told is a somewhat common style there (usually just the bass, the melody holds over 3.) The conscious effort to do this brings back some of the "pat head while rubbing belly" two hands difficulty, and I find it also a bit of a struggle to consistently play the chord button on 3. I think that's the style, but my left hand often wants to go to counter-bass or alternate-bass on 3. I'm sure it will become natural if I play the tune more, it's just this conscious stage that's naturally a little rough.
Another Norwegian 3 style, by, the way, takes extra time for 2; I don't think those two features are found together in the same tune though.