For an extra hint of sweetness, drizzle the soup with a bit of honey just before serving.
Peel, halve, and seed 2 cucumbers, then coarsely chop. (You should have about 4 1/2 cups.) Halve and seed remaining (unpeeled) cucumber; cut into a 1/4-inch dice. Working in batches, puree chopped cucumbers, melon, 1/2 cup lime juice, mint, and ice in a food processor or blender. Season with salt and more lime juice. Transfer soup to a bowl, cover, and refrigerate until ready to serve, up to 2 hours.
Smear 1/4 cup yogurt onto two sides of each serving bowl. Spoon chilled soup into bowls, drizzle with oil, and sprinkle with diced cucumbers and mint. Serve, with lime wedges.