Cook potatoes in large pot of boiling salted water until tender, about 30 minutes. Drain. Cool potatoes completely. Peel potatoes; quarter lengthwise. Cut crosswise into 1/2-inch-thick slices. Place potatoes in large bowl; sprinkle generously with coarse salt and pepper. Add drained cucumbers, onion, sliced radishes, and remaining 3 tablespoons dill; toss to blend. Let stand 1 hour. Stir mayonnaise into salad. Season generously with salt and pepper, if desired. (Salad can be made 1 day ahead. Cover and refrigerate.)
04Mound salad in bowl; garnish with whole radishes. Serve cold or at room temperature.