Place a saucepan filled 2/3 with water over high heat. Cut the potatoes into 1-inch pieces. Add the potatoes to pan; cover and bring to a boil. Reduce the heat to medium-high and cook until tender, about 5 minutes. Drain.
Combine the cucumber, sour cream, yogurt, dill, salt, and pepper in a large bowl. Add the drained potatoes to the cucumber mixture, and toss gently to coat.