Drain the tuna in a colander and press out as much oil as possible with a spoon. Rinse the cherry tomatoes, cucumber, and yellow bell pepper. Halve the tomatoes, slice the cucumber into thin slices, and cut the bell pepper (without the seeds) into strips. Peel and mince the red onion as finely as possible.
Place the drained tuna in a large mixing bowl and break it up with a fork into large chunks. Add the prepared cherry tomatoes, cucumber, bell pepper, and red onion to the bowl. Gently toss the ingredients together.
In a separate small bowl, stir together the mayonnaise, freshly squeezed lemon juice, salt, and freshly ground black pepper until you have a smooth dressing. Pour this sauce over the vegetable-tuna mixture and toss everything together well so all ingredients are coated with the dressing.
Distribute the mixed greens over the plates or place them on a shallow serving dish. Spoon the prepared tuna salad on top of the greens and serve immediately.