Traveling in shoulder season and off-season, when fewer tourists frequent favorite destinations, will almost always save you money. Case in point: this round-trip fare from LAX to Athens.
From Sept. 26-Dec. 21 and Jan. 21-Feb. 27, you can depart for Athens for $759 on Aeroflot.
Availability may be limited. You must stay at least a Saturday night, but you can depart any day of the week. Travel must be completed by March 2015.