With a knife, slice the watermelon into inch-thick slabs and remove the rind. Cut into chunks and put into large bowl.
Remove the rind from the jicama and cut into 1/2-inch thick batons. Details: with the knife, cut a slab off one end of the jicama and place this side down on your cutting board to stabilize. Remove the remaining rind by cutting down all the way around. Cut off other end so rind is completely removed. Now cut the jicama in half then into planks about 1/2-inch thick. Flip the planks on their side and cut into 1/2-inch thick batons. Add to the watermelon.
Cut lime in half and squeeze over the watermelon and jicama. Drizzle with the olive oil. Pour out onto a serving plate or bowl. Tear basil and/or mint leaves and scatter over the top. Sprinkle with salt, and optional cheese. Arrange arugula or other greens on the side and serve.