Place the cucumber slices in a small bowl and sprinkle on 1/2 teaspoon of salt. Toss and let sit for 20-30 minutes. (Salting the cucumber releases moisture and prevents them from turning mushy and the salad from becoming runny.)
Wash and scrub the potatoes. Leave skins on and quarter each potato. Add the potatoes and carrots to a large pot and add enough water to cover potatoes. Gently boil until the potatoes are cooked through, about 10-12 minutes.
Remove potatoes and carrots and place onto a cutting board to cool. The potato skins may peel off with little effort. If not, use a knife to remove the skins. Dice potatoes and carrots and place in a large mixing bowl.
Finely chop the hard-boiled eggs and add to the mixing bowl.
With your hands or a towel, firmly squeeze the liquid from the cucumbers. (I was able to squeeze out 1/4 cup of liquid.) Add the cucumbers to the mixing bowl. Discard the liquid.
Add mayonnaise and salt and pepper to taste. Combine all of the ingredients (mash some of the potato cubes with a fork for a creamier texture). Refrigerate the salad for about one hour before serving.