Combine sugar and 1 cup water in a small saucepan. Heat over medium heat, stirring until dissolved. Remove from heat and let cool.
Wash and roughly chop the cucumbers. Blend with a splash of water until smooth and bright green.
Pour the cucumber puree through a fine mesh strainer. Press gently with a spoon to extract smooth cucumber juice.
Roll the lemons on the counter, cut, and squeeze until you have 1 cup fresh lemon juice. Remove any seeds.
In a large pitcher, combine cucumber juice, lemon juice, remaining water, and cooled syrup. Stir well and adjust sweetness or tartness to taste.
Refrigerate for at least 30 minutes. Serve over ice with cucumber slices, lemon wheels, or fresh mint.
