When my wife and I were married, our priest put the rings on our left hands. We just went with it, though afterwards (like, a year or so) I brought up the idea of changing. She didn't want to at first so, like Gabriel so sagely said, I didn't push it. I, however, switched my own ring to the right hand.
I remember, though, some time later when wifey and I were sitting in a restaurant, and I noticed that she was smiling at me strangely, as if something amused her. "What?" I asked.
"You haven't noticed it yet!" she said sweetly.
Oh, great, I thought. Hair? Clothes? What?
Then she held up her right hand, right in my face (Yeah, I'm pretty clueless), with the ring on it. Been there ever since. Most folks here at SVS that I've noticed have them on the right hand, but the left-hand custom still seems to be the norm for Anglo societies.
Ultimately, wearing it on the right hand is a nice, minor image (or "icon," as a priest liked to tell me) of the significance of marriage to an Orthodox Christian couple, but neglecting it won't be the end of the world. Much better to let one's whole married life be that icon.