I actually wouldn't do that unless your heart is set on being in Athens. It will probably be easier to find award inventory USA-(Somewhere in Europe where you can buy a cheap DE or other LCC ticket)-JTR-(somewhere else in Europe using an LCC to get there from JTR)-USA as opposed to forcing travel through ATH. If you're going to have to buy positioning flights once you are in Europe anyway, there's not THAT much advantage to flying intra-European J from AMS or CDG on an AS award. Plus bonus Europe!

FWIW, BA YQ and UK APD comes off quite substantially ex-DUB, AMS or INV.