In a large bowl, combine raspberries and sugar. Mash with a muddler or fork until the berries release their juices.
Stir in lemon juice and 1.0 cup of the water. Let sit for 2 minutes to help the sugar dissolve.
Pour the mixture through a fine-mesh strainer into a pitcher, pressing to extract as much liquid as possible. Discard seeds and pulp.
Add the remaining 2.0 cups water and the ice. Stir until chilled, then taste and adjust sweetness or tartness with a little more sugar or water if needed.
Pour into glasses, add lemon slices and mint for garnish, and serve immediately.
Notes
For a sparkling twist, replace 1.0 cup of water with chilled club soda just before serving. For a slushy version, blend with 2.0 cups ice.