I wore my wedding ring on my right hand for a while, out of convenience. It didn't get in the way of the strap (which can be uncomfortable, at least with my hands and ring) and wasn't exactly countercultural since European men wear their wedding rings on their right hands. Eventually I was made aware that it was really bothering my mother in law, and decided it wasn't worth the argument. Now I just move it to the other hand when I put on tefillin.

I've thought quite a bit about the symbology of having the ring on one hand and the tefillah on the other. Not that I've come to any conclusions, mind you.