Peel cucumbers into a striped pattern lengthwise using a vegetable peeler. Slice cucumbers in half lengthwise; cut each half into 1/4-inch slices creating 'half-moon' shapes. Combine cucumbers, watermelon, and red onion in a large bowl.
Whisk rice vinegar, white wine vinegar, and lime juice together in a bowl; add mint. Muddle the mint by vigorously stirring and pressing the leaves with a spoon until they look bruised but aren't falling apart. Remove and discard mint leaves.
Drizzle the dressing over salad and gently toss. Sprinkle feta cheese over salad; toss gently.
