In a large saucepan, bring 3 cups of the water to a boil. Remove from heat. Add tea bags and mint; steep 6 minutes.
Remove tea bags. Leave mint in tea, pressing stems and leaves against pan with the back of a spoon to extract flavor. Stir in 3 cups cold water and apricot nectar. Chill, covered, for 4 hours.
Remove and discard mint. Serve over ice, garnished with mint sprigs and/or apricot wedges, if desired.