Place the cucumber and mint in a blender and blend until smooth.
Place a sieve over a bowl and strain the cucumber and mint puree into the bowl.
Transfer the puree from the bowl into a serving jug.
Add the lemon juice and agave syrup to the jug, and stir until all the liquid is completely combined. Then pour in the water and stir.
Add ice to the jug and decorate with slices of lemon and fresh mint leaves.