In a bowl, combine the chicken cubes with 3 tablespoons olive oil, lemon juice, 2 minced garlic cloves, oregano, 1 teaspoon salt, and black pepper. Toss to coat and let sit for 10 minutes while you prepare the rest.
In a separate bowl, stir together the Greek yogurt, grated cucumber, remaining minced garlic clove, and fresh dill. Cover and refrigerate until serving.
Bring 1 1/4 cups water to a boil in a small saucepan. Stir in the couscous, 1 tablespoon olive oil, and a pinch of salt. Cover, remove from heat, and let stand for 5 minutes.
Thread the marinated chicken onto skewers. Grill over medium-high heat for 8 to 10 minutes, turning once, until the chicken is cooked through and lightly charred.
Fluff the couscous with a fork. Serve the grilled chicken skewers over the couscous with a generous spoonful of tzatziki on the side.