In a medium saucepan, combine fruit, 3 cups of water, sugar, lemon juice, reserved lemon halves, and cinnamon sticks. Bring to a boil, and simmer for 30 minutes. Remove from heat, and let cool to room temperature. Remove cinnamon sticks and lemon halves, and coarsely chop by hand or in a food processor. If using a food processor, you may add the cooked lemon halves if desired.
May be stored in an airtight container in the refrigerator for up to 1 week