To marinate the chicken: In a non-reactive dish, combine the lemon juice, olive oil, oregano, salt, and pepper and mix together. Add the chicken breasts to the dish and rub both sides in the mixture. Cover the dish with plastic wrap and let marinate in the refrigerator for at least 30 minutes and up to 4 hours.
To make the dressing: Combine all the ingredients in a resealable container and shake vigorously. Refrigerate until ready to serve, and then bring to room temperature before tossing salad.
Assemble the salad: cut off any dark tips and the bitter white bottoms from the romaine leaves. Cut the lettuce into 1-inch strips and place in a bowl that is large enough to hold all the salad ingredients comfortably. Scatter cucumbers, tomatoes, red onion, olives, and feta over the top. You can prepare the salad up to a few hours in advance. Cover it with a moist paper towel and refrigerate until 30 minutes before serving.
To cook the chicken: Heat a nonstick skillet or grill pan over high heat. Add the chicken breasts and cook, turning once, until well browned, about 4 to 5 minutes on each side or until cooked through. Let the chicken rest on a cutting board for a few minutes before slicing it into thin strips.
Give the dressing a good shake and pour it into a nice little serving bowl, using the lid to strain out the garlic.
Toss salad just before serving and fan chicken out on top.