To make the simple syrup, add the granulated sugar and 1 cup of water into a small saucepan. Heat over medium heat, stirring occasionally, until the sugar completely dissolves.
Remove the simple syrup from the heat and let it cool completely, for at least 30 minutes.
Add the cucumber chunks to a blender. Blend until smooth.
Place a fine-mesh sieve over a bowl or pitcher and pour the cucumber puree through the sieve. Use the back of a spatula to press out as much liquid as possible. You should get about 1 cup (8 oz) of cucumber juice. Discard any solids left in the fine-mesh sieve.
Add the simple syrup, cucumber juice and lemon juice to a large pitcher.
Add the remaining 3-4 cups of water to the pitcher. Start with 3 cups, taste the lemonade, and then add up to an additional 1 cup of water if the concentrate still needs to be diluted. If the cucumber lemonade is too watered down, add another 2-4 tablespoons of lemon juice from 1 lemon.
Stir well to combine.
Chill the cucumber lemonade in the refrigerator for at least an hour before serving.
Serve the lemonade chilled in glasses of ice. Garnish with lemon slices, cucumber ribbons, and fresh mint as desired.