In a small bowl, stir together the yogurt, dill and cumin. Season to taste with lemon juice, salt and pepper.
Cook the burgers on a hot grill or in a hot cast-iron skillet, to 165° F as measured by a meat thermometer.
To serve, arrange the cucumbers, tomatoes and slices of feta on the bottom hamburger buns. Top with the cooked burgers. Spread the yogurt sauce evenly over the burgers and cover with the top buns. Serve at once.