Place the beets into a large saucepan with enough water to cover the beets by 1 inch. Bring to a boil. Reduce heat to medium-low, cover, and simmer until the beets are easily pierced with a fork, about 40 minutes; drain. Chill beets in refrigerator until cool enough to handle.
Peel the beets and grate into a large bowl. Stir the prunes, walnuts, and pickle with the beets. Add the mayonnaise and stir until all the ingredients are evenly coated. Chill completely before serving.
