Assalamu Alaikum wa Rahmatullah
according to narrations we have many advice regarding wearing the rings on the right hand, one of which is the famous hadith in which we read about the Ziarat of arbaeen.
These recommendations are apparently for the recommended stones like Aqeeq and those on which there is a Zikr.
On the other hand we have disrecommendations regarding wearing them on the left hand which you know is usually used for Istinjaa(purifying the body)
So as a conclusion we can say that there is no recommendation or disrecommendation with regards to the marriage rings nowadays unless they have recommended stones on them like Aqeeq of Yemen or they have Zikr on them.