When a wave travels in a material from one point to another, it propagates in compliance with the principle of reciprocity: if the locations of the source and receiver are interchanged, the recorded signal remains unchanged. This symmetry property of wave propagation has served as a powerful modeling and experimental tool in vibroacoustics for over a century. In a recent change of perspective, reciprocity is regarded as a hindrance for creation of intrinsically biased waveguides and materials, such as one-way mirrors and diodes for elastic waves. Recent experiments have shown the feasibility of creating such devices in the context of lattice materials and mechanical metamaterials. In this light, we review the scenarios for realizing nonreciprocal dynamics in periodic elastic media, with a focus on strategies that utilize nonlinearity for achieving this.