Etihad equity partner airline Air Serbia now flies New York JFK – Belgrade 5 times weekly. And you can book their service with connections to Sofia, to Sarajevo, to Bucharest as well as several other cities for $215 roundtrip.
Outbound travel is permitted August 21 through April 1, and a Saturday night stay is required. Fare rules say this must be ticketed by Thursday, but is likely not to last that long.