Combine peach pieces, sugar, and water in a medium saucepan. Bring to a boil over medium heat, mashing the peaches against the side of the pan. Simmer for 5 minutes, then remove from heat and cover for 30 minutes.

For clear syrup, pour the mixture through a fine-mesh strainer into a jar or other storage container. If desired, you can add a little of the fruit back into the syrup. For thicker syrup, use a blender or immersion blender to blend the mixture.