Marinate the Chicken: In a bowl, combine olive oil, lemon juice, oregano, garlic, salt and pepper. Add chicken; let marinate at least 15 minutes (or up to 24 hours, in the refrigerator for a stronger flavor).
Grill the Chicken: Light a grill or heat a grill pan or skillet over medium heat. Cook 6-7 minutes on each side or until the chicken is 165°F inside. Take off heat and allow to rest before slicing.
Prepare the Salad:
In a large bowl, toss together chopped cucumber, cherry tomatoes, red onion, olives, and feta cheese.
Make the Dressing: Whisk red wine vinegar with olive oil, oregano, salt and pepper. Drizzle over the salad and toss to coat lightly.
Assemble: Cut the grilled chicken into slices and place it on top of the salad. Garnish with more fresh parsley if desired.