Roughly chop the cucumbers, bell peppers, tomatoes, and red onions into 1-inch cubes. Put each vegetable separately into a food processor fitted with a steel blade and pulse until it is coarsely chopped. Do not overprocess!
After each vegetable is processed, combine them in a large bowl and add the garlic, tomato juice, vinegar, olive oil, salt, and pepper. Mix well and chill before serving. The longer gazpacho sits, the more the flavors develop.