Mince 1/2 cup green pepper; set aside for garnish. Place remaining pepper in large bowl. Cut cucumber lengthwise in half. With spoon, remove and discard seeds. Coarsely chop cucumber. Mince 1/4 cup cucumber; set aside for garnish. Add remaining cucumber to pepper in large bowl.
Cut avocado into chunks. In large bowl, add avocado, tomatoes, cilantro, onion, garlic, lime juice, olive oil, sugar, cumin, salt and cayenne to cucumbers and pepper; toss. In batches, puree vegetable mixture in food processor until smooth. Transfer to another large bowl or 8-cup measuring cup. Stir in 1-1/2 cups cold water; cover and refrigerate 2 hours to allow flavors to blend and soup to chill.
Ladle soup into bowls, garnish with additional cilantro leaves, minced green pepper and cucumber. Serve chilled.