I think Bonnie makes a good point. Tom bought a knock off Rolex in (of course) Shanghi but I don't think he should wear it in Rio...it looks exactly like the real thing. We've never had any type mugging experience, don't need to start now!
I always wear my wedding ring, and depending on where I am, on occassion I twirl it inside so the diamond is hidden. I tend to do this in very poor countries (but will in Rio), not so much for fear of a mugging or anything, I just don't feel comfortable coming across as "flaunting".