A multidimensional semiclassical method is developed for calculating energy-level splittings in polyatomic molecules due to multiple tunneling motions. The procedure can be easily implemented within standard trajectory simulations and thus allows for full-dimensional treatment of tunneling in large systems. The method is applied to alpha-methyl-beta-hydroxyacrolein and the calculated ground-state splitting is in good agreement with the experimental value. This study further demonstrates that facile semiclassical approaches can be used to accurately treat tunneling in many-atom systems.