Bishops should wear rings; most of them opt for the plain gold band that JP II wears.
I don't wear a wedding ring. I've got some strange (and rare) allergic reaction to rings or watches made with precious metal. I've never been able to wear jewelry, and have to wear a cheap watch with a leather band.
No medals around the neck, either. The brown scapular is all I can wear, which I've worn since I was in grade school.
Mary made that promise, you know, to those who wear her scapular, and I'm holding her to it.