Propagation of a Gaussian beam in a layered periodic structure is studied analytically, numerically, and experimentally. It is demonstrated that for a special set of parameters the acoustic beam propagates without diffraction spreading. This propagation is also accompanied by negative refraction of the direction of phase velocity of the Bloch wave. In the study of two-dimensional viscous phononic crystals with asymmetrical solid inclusions, it was discovered that acoustic transmission is nonreciprocal. The effect of nonreciprocity in a static viscous environment is due to broken PT symmetry of the system as a whole. The difference in transmission is caused by the asymmetrical transmission and dissipation. The asymmetrical transmission is caused solely by broken mirror symmetry and could appear even in a lossless system. Asymmetrical dissipation of sound is a time-irreversible phenomenon that arises only if both energy dissipation and broken parity symmetry are present in the system. The numerical results for both types of phononic crystals were verified experimentally. Proposed devices could be exploited as collimation, rectification, and isolation acoustic devices.