In a small bowl, whisk together the extra virgin olive oil, half of the lemon juice, minced garlic, dried oregano, sea salt, and black pepper.
Place the chicken breast in a shallow dish and coat thoroughly with the marinade; let it sit for at least 15 minutes for the flavors to penetrate.
Heat a grill or grill pan over medium-high heat and cook the chicken for 6-7 minutes per side until the internal temperature reaches 165°F.
While the chicken cooks, prepare the tzatziki by mixing the non-fat Greek yogurt with the fresh dill and the remaining lemon juice.
Chop the cucumber, cherry tomatoes, and red onion, then toss them together in a small bowl.
Slice the grilled chicken into thin strips.
Assemble the bowl by placing the cooked brown rice at the base, then topping with the sliced chicken, cucumber salad, crumbled feta cheese, and a dollop of the dill tzatziki.