Whisk together the olive oil, lemon juice, minced garlic, oregano, salt, and pepper in a small bowl to create the vinaigrette/marinade.
Place chicken breasts between plastic wrap and pound to a uniform thickness of 0.75 inches for even cooking.
Marinate chicken in 3 tablespoons of the prepared vinaigrette for 15 minutes at room temperature.
Preheat grill or grill pan to medium-high heat (approximately 400 degrees Fahrenheit) and lightly coat with neutral oil.
Grill chicken for 6 to 7 minutes per side until the internal temperature reaches 165 degrees Fahrenheit (74 degrees Celsius).
Transfer chicken to a cutting board and allow to rest for 5 minutes before slicing into 0.5-inch thick strips.
In a large chilled bowl, combine mixed greens, tomatoes, cucumber, and red onion.
Toss the vegetable mixture with the remaining vinaigrette until evenly coated.
Divide the salad base among four plates and top each with a portion of the sliced grilled chicken.