To cube the cucumber: Cut the ends off the cucumber. Then thinly slice off all four long sides of the cucumber so that it resembles a long rectangle. Cut into thirteen 1-inch cubes. (Reserve the remaining cucumber for another use.)
To cube the watermelon: Cut off the ends. Cut the watermelon into 1-inch round slices. Use a cucumber cube as a guide to cut the watermelon into thirteen 1-inch cubes. (Discard the rind and reserve the remaining watermelon for another use.)
To cube the feta: Cut into four 1-inch cubes using a cucumber cube as a guide.
To build the salad, make the bottom layer: Arrange 3 cubes across the top of a serving platter and then 5 cubes down, alternating cucumber and watermelon cubes (you should have something that resembles the number "7"). Fill in the "7" to make a 3-by-5-inch rectangle by alternating cucumber and watermelon cubes to create a checkerboard pattern. Occasionally swap out a cube of cucumber or watermelon with a cube of feta to integrate the 4 cubes into the pattern. There should be 15 cubes. To build the top layer, repeat the checkerboard pattern with the remaining 15 cucumber and watermelon cubes making sure that all the feta cubes are covered and every bottom cucumber cube is topped with a watermelon cube and vice versa.
Drizzle the cubed salad with the oil and then sprinkle with the pepper, basil and mint leaves.