Marinate the chicken: In a bowl, combine olive oil, lemon juice, oregano, garlic, salt, and pepper. Add chicken and toss to coat.
Marinate 20–30 minutes (or up to 4 hours in the fridge).
Make the dressing: Whisk olive oil, red wine vinegar, lemon juice, Dijon, oregano, garlic, salt, and pepper until emulsified. Taste and adjust salt, acid, or pepper as needed.
Prep the vegetables: Slice the cucumber, halve the tomatoes, thinly slice the red onion, and chop the herbs. If using greens, chop them into bite-size pieces.
Cook the chicken: Heat a grill pan or skillet over medium-high.
Cook chicken 5–6 minutes per side, or until the internal temperature reaches 165°F/74°C. Rest 5 minutes, then slice or cube.
Assemble the salad base: In a large bowl, add cucumber, tomatoes, red onion, olives, and herbs. If using greens, add them now.
Add cheese and chicken: Gently fold in feta and the sliced chicken.
Dress and toss: Drizzle on about two-thirds of the dressing.
Toss lightly to coat. Add more dressing as needed, a little at a time, to avoid overdressing.
Finish and serve: Taste and adjust with a pinch of salt, extra lemon, or a grind of pepper. Serve right away, or chill briefly to let flavors mingle.