The decay of Alfvén waves in a filamentary molecular cloud is investigated through three-dimensional numerical simulations. We have considered a filamentary molecular cloud supported in part by the Alfvén wave against the self-gravity. Our attention has been focused on the basic physical mechanism for the decay. The decay rate is obtained as a function of the wavelength and amplitude. It is found that when the wave is circularly polarized, the decay e-folding timescale is several times the fast wave crossing timescale for the filament and independent of the wavelength, whereas when the wave is linearly polarized, the amplitude of the wave decreases inversely proportional to time. It is also found that the decay of Alfvén waves induces rotation and shear flow in the filamentary cloud. The propagation of two Alfvén waves in the medium results in the excitation of daughter waves due to nonlinear coupling between mother waves. The wavenumber of the daughter waves is the sum or difference between those of the mother waves, and below a critical wavenumber of the daughter wave, the filamentary cloud fragments as a result of Jeans instability. The fragments collapse to form high-density rotating magnetized disks. In contrast, below a critical wavenumber of the mother wave, the cloud becomes a dense helical filament after the decay of the Alfvén waves. The present models are compared with previous simulations and observations with regard to the rotation, fragmentation, and helical structure of filamentary clouds.