@Marie

Is your impression that the EO Church is growing, dying, or just maintaining in the US?

Because that seems like the polite thing to do - introduce myself as a visitor to the parish if the priest was available before Mass.  My assumption is that it would come up during the introduction.

I also didn't know that there was a Byzantine Rite parish in my area.  Just searched and found that there is one small parish that I didn't know about.