Engagement rings go on the left hand, if they are given. There is no Orthodox tradition or reason to wear one on the right--actually,the opposite is the case. Ive seen some people start to wear one on the right in the mistaken idea that they are being "Orthodox" in doing so. However, wearing a separate engagement ring doesnt have a very long history--a couple of hundred years, perhaps, and almost no history in Orthodox countries, and wearing an engagement ring on the right hand has no history in Orthodoxy.
Instead, in countries such as Greece, one was given a thin gold band upon their engagement. Both men and women had such a ring. Then, upon their wedding, during the betrothal part, the priest would move that same ring to the right hand. There was only one ring, and as you can see, there was actually a reason to wear it on the LEFT hand before it was blessed by the priest and moved to the right.
In the end, it's not that ring that matters but the wedding ring, so there's no real importance to any hand or finger you wear it on. As long as you are aware there's no "Orthodox" reason to wear it on the right, wear it wherever you want!