We report extraordinary effects in the transmission of sound through periodically perforated plates, supported by both measurements and theory. In agreement with recent observations in slit arrays, M. H. Lu et al. [Phys. Rev. Lett. 99, 174301 (2007)10.1103/PhysRevLett.99.174301], nearly full transmission is observed at certain resonant frequencies, pointing out similarities of the acoustic phenomena and their optical counterpart. However, acoustic screening well beyond that predicted by the mass law is achieved over a wide range of wavelengths in the vicinity of the period of the array, resulting in fundamentally unique behavior of the sound as compared to light.