Combine 2 cups apple juice, lime, and ginger in a small saucepan. Bring to a boil; reduce heat and simmer, uncovered, 3 minutes. Remove from heat; cover and let cool.
Strain juice mixture into a pitcher; discard ginger and lime. Stir in remaining 2 cups apple juice, pineapple juice, and orange juice; chill thoroughly.
Note: For a Rum-Spiked Cooler, mix 4 parts Gingered Citrus Cooler with 1 part golden rum.