Leave roots and 1 inch of stems on beets; scrub well with a vegetable brush. Place in a saucepan; add water to cover. Bring to a boil; cover, reduce heat, and simmer for 35 minutes or until tender. Drain; rinse under cold running water. Drain; let cool. Trim off beet stems and roots; rub off skins. Cut the beets into 1/4-inch-wide strips.
Arrange beets, cabbage, carrot, cucumber, and onion on a serving platter. Pour Lemon Vinaigrette over salad.