In June, after ten days exploring Greece, we would enjoy a three-day destination in another country? We are looking for a quick round-trip visit worth doing. Thanks.
You'll have to fly. Even then you'll spend the better part of a day moving in both directions, leaving you with only 1 full day at your destination. It's not worth it. Figure out somewhere else in Greece to visit instead.
Agree with Lee. There is really nowhere near Greece that might be easy to get to, other than maybe Istanbul. Greece is not well served by public transportation and is pretty hard to get back and forth with its neighbors. You could look into flying into Istanbul. Less than 3 days in Israel doesn't hardly seem worth it but if you just visited Jerusalem, maybe...
If your flight home from Greece makes a connection in Paris, Amsterdam, or London, then that could be a natural spot to build in a stop-over. A multi-city, 3-leg trip with stopover may cost more than a multi-city, 2-leg trip (such into Athens and home from Paris), in which case you'd also compare the cost of buying the (proposed) Athens-Paris ticket separately; see www.skyscanner.com for budget flights.