Cook potatoes: Place potatoes in a large pot of cool, salted water over high heat. Bring to a boil, then reduce to medium-high and allow to boil until the potatoes are fork-tender, about 10-12 minutes. Drain and allow to cool completely, if possible.
Make the dressing: In a small bowl, whisk together Greek yogurt, lemon juice, garlic, dill and salt.
Place the cool potatoes in a large bowl and pour the yogurt mixture over the top. Mix gently to coat -- be careful not to break the potatoes up too much.
Gently fold in the chopped cucumber, olives and feta cheese.
Refrigerate for 3-4 hours before serving, if possible, to allow the flavors to meld.