In a small bowl, whisk together 0.5 tbsp olive oil, lemon juice, minced garlic, and half of the dried oregano.
Place the chicken breast in a shallow dish, coat with the marinade, and let sit for 15 minutes.
Preheat a grill or grill pan over medium-high heat.
While the grill heats, chop the cucumber, halve the cherry tomatoes, and finely dice the red onion.
In a medium bowl, toss the vegetables with the remaining 0.5 tbsp olive oil, remaining dried oregano, fresh parsley, sea salt, and black pepper.
Grill the chicken for 6-7 minutes per side or until the internal temperature reaches 165°F.
Let the chicken rest for 5 minutes before slicing and serving alongside the fresh salad.