Mix lemon juice, olive oil, minced garlic, oregano, salt, and pepper in a bowl. Coat the chicken breasts well, cover, and let them marinate for at least 30 minutes to absorb all flavors.
Grate the cucumber, squeeze out excess water, then combine it with Greek yogurt, dill, garlic, lemon juice, and a pinch of salt. Chill in the fridge while the chicken cooks to let flavors meld.
Cut marinated chicken into bite-size pieces and thread onto skewers. Grill over medium heat for 10–12 minutes, turning occasionally, until chicken is golden and cooked through.
Plate the chicken skewers with a generous dollop of tzatziki. Garnish with extra dill or a wedge of lemon for a bright, restaurant-style presentation.