Whisk together olive oil, lemon juice, oregano, garlic powder, salt, and pepper in a bowl to create the marinade.
Coat the chicken breasts thoroughly with the marinade and refrigerate for 15 minutes.
Preheat a gas or charcoal grill to medium-high heat, approximately 400°F (204°C).
Place chicken on the grill and cook for 6 to 7 minutes per side, ensuring the internal temperature reaches 165°F (74°C).
Transfer chicken to a cutting board and allow it to rest for 5 minutes to retain juices before slicing into 1/2-inch strips.
Combine mixed greens, tomatoes, cucumber, and red onion in a large mixing bowl.
Add the balsamic vinaigrette to the vegetables and toss gently to coat.
Apportion the salad mixture onto four plates and top each with a sliced chicken breast and 1 tablespoon of feta cheese.