To make the apple juice ice cubes, pour the apple juice into two ice trays and freeze until the sangria is ready to serve.
Combine the water, mint leaves, sugar, and cinnamon in a small saucepan, and bring to a boil over medium heat. Reduce the heat and simmer for several minutes. Remove from the heat and allow to cool. Once the mixture has cooled to room temperature, remove and discard the mint and cinnamon sticks.
Transfer the remaining mixture to a large serving bowl.
Add the grape juice, peaches, pears, and the orange and lemon slices to the serving bowl. Mix well, and refrigerate overnight. Immediately before serving, mix in the sparkling apple cider and the apple juice ice cubes.