Store leftover sandwich filling in the refrigerator up to 3 days.
Peel and seed cucumber; dice cucumber and bell pepper. Grate carrots, and mince garlic.
Combine cucumber, bell pepper, carrot, garlic, green onions, and next 4 ingredients in a large bowl, stirring until well blended. Spread mixture on half of rye bread slices, and top with remaining slices. Cut vegetable sandwiches in half diagonally.