My wife and I did not exchange rings in the ceremony. We had bible bearers that walked down a bible that had special meaning to her and one that had special meaning to me. I do not own a band but I did buy my wife a simple, inexpensive band to wear on our honeymoon cruise because she wanted to have something. I am fine with that. She does not wear it anywhere now.
Why I used this reference.
Mark 9:42
And whosoever shall offend one of these little ones that believe in me, it is better for him that a millstone were hanged about his neck, and he were cast into the sea.
Paul said to be all things to all men. It is better for me not to wear a ring and allow a larger circle of fellowship than wear a ring and offend the more conservative brethren.
Just me opinion on the matter