In a blender, combine the cucumber, tequila, lime and simple syrup. Blend until the mixture is as smooth as you can get it. In a cocktail shaker, combine half of the mixture with 8 to 10 ice cubes, cover and shake for a full 15 seconds. Strain through the shaker's top into 2 martini glasses.
Garnish with cucumber slices (I usually cut a slit in each slice and impale one on the side of each glass). Repeat the shaking with the remaining margarita mix.