Grate carrots or cut into matchsticks. Peel cucumber, slice in half, and remove seeds. Cut into matchsticks or into thin pieces that are approximately 1 1/2 inches long.
Lay out sushi wrappers. Follow package instructions, if there are any. (Sometimes wrappers require placing into warm water to make them more pliable.) Add an even amount of rice, spinach, tuna, carrots, and cucumbers to each wrap and roll up tightly. Cut into quarters.