Cook the rice according to the package directions.
Meanwhile, thread the lamb onto eight 6-inch skewers. Season with the oregano, ½ teaspoon salt, and ¼ teaspoon black pepper.
Heat grill or grill pan to medium-high. Cook the lamb skewers, turning occasionally, to the desired doneness, 8 to 10 minutes for medium-rare.
Meanwhile, in a small bowl, combine the cucumber, Feta, onion, dill, oil, lemon juice, and ¼ teaspoon each salt and pepper.
Serve the kebabs with the rice and relish.