My priest (and this is a fairly large congregation) will often ask: "Do we have any visitors here today?"
The people who are visiting raise their hands, and do not stand up.
He then asks them, "Where are you visting from?"
They answer, and that is it...
Ofcourse, you don't have to raise your hand, so it affords a way out for someone who is not Orthodox, or is too shy.
Unfortunately, I have noticed that priests are often too busy when giving out andidoron to notice too many things...though when visiting churches, this is a good time to say hello to the priest and quickly introduce yourself, or comment on something you liked.
Also, atleast in some Greek Orthodox churches, the priests often seem to get busy (with what in particular--I don't know) after all have gone to the hall for coffee, so one usually doesn't see them there until almost all have left. (Because politics are rampant in Greek churches, it may be that the parish priest does not want to socialize there for that very reason--who knows?!?)
Coffee time (or 'agape') as the Greeks call it, is a good time to meet people, but I find when visiting other parishes, that in Greek churches, (and even in larger churches of other jurisdictions) you may just be standing there, alone or with your spouse, drinking coffee alone. Rarely, it seems, do people take the time to come and say hello...

In Christ,
Alice