Place the cucumbers and lime juice in a food processor and process until no big pieces remain. (Alternatively, add the freshly squeezed lime juice and about 2 cups of the cucumbers to a blender and blend together to liquefy the cucumber. Then, add the remaining cucumbers and pulse until no large pieces remain and you have a thick liquid.)
Transfer the mixture to a large bowl and fold in the jalapeños, red peppers, and mint. (To chiffonade the mint leaves: Lay the leaves one on top of the other, and roll tightly. Using a sharp knife, slice the roll very thin, so you have thin strips of mint. Doing it this way helps keep the herbs from bruising under the pressure of the knife.)
Season to taste with salt and pepper.
Keep the Cucumber-Mint Gazpacho chilled in the refrigerator until ready to serve. Divide among bowls and garnish with mint leaves.