Combine lemon juice, brown sugar, olive oil, garlic, and oregano.
Mix well. Can be made earlier in the day and refrigerated, and re-mixed prior to adding to the chicken.
Pierce chicken beasts with a fork twice on each side.
Combine chicken and marinade and let sit at room temperature (or refrigerated) for 20 - 30 minutes.
Preheat BBQ or Oven to 400 degrees. Cook on BBQ 8 minutes each side, or in oven approximately the same on a broiler pan to allow any fat or marinade to escape (time for both BBQ and Oven can differ from these times (i.e. BBQ - outside temp can affect the internal heat of the BBQ. General rule is to cook until juices run clear.
***This can also be made on an indoor grill if you have one. I've done that and it turned out great. You can probably broil it but I've never tried.