Trim all fat and tendons from chicken breasts and trim underneath side if needed so chicken pieces are all the same thickness.
Make small slits crosswise going down each chicken breast to help the marinade penetrate.
Use a chef’s knife to finely chop capers.
Crush the oregano with a rolling pin or meat mallet.
Mix together olive oil, lemon juice, oregano, garlic puree, caper juice, chopped capers, and black pepper. (You could also put all the marinade ingredients in a blender and process them rather than chopping the capers and crushing the oregano.)
Lay chicken flat in a large Ziploc bag and pour marinade over. Let chicken marinate in the refrigerator, preferably 4-6 hours or longer.
When you're ready to cook, drain chicken well in a colander placed in the sink.
Spray the grill grates with non-stick spray, then preheat gas or charcoal grill to medium-high. (You can only hold your hand there for a few seconds at that temperature.)
Lay the chicken pieces at an angle to the grill grates and cook for 3-4 minutes without moving.
Then lift the edge of one piece of chicken to see if grill marks are as dark as you’d like them.
When they look good, rotate the chicken to the opposite angle and cook 3-4 minutes more on the first side.
When you have good grill marks, turn chicken to the opposite side and cook 2-3 minutes more (or longer for very thick chicken breasts.)
Chicken is done when it feels firm (but not hard) to the touch when you press down on it with your finger.
Don’t overcook. Serve hot.
This chicken could also be cooked in a stove-top grill pan with ridges or a George Foreman Grill with good results.
