Recent advances in the fabrication of nanoscale material systems have made it possible to alter precisely the atomic structure in ways that enhance the properties and allow for certain functions to be realized. This work is concerned with two-dimensional periodic structures and emphasizes the effects of intentional defects on their wave propagation characteristics. In this draft paper, investigations are limited to a two-dimensional spring-mass lattice, composed of “super-cells” where mass inclusions are added to alter band-gap properties, as well as low frequency directionality. The presented results will then be extended to two-dimensional nanostructures, such as graphene nanosheets, in order to investigate their application as nanoscale acoustic waveguides, where engineered defects, uniformally distributed across the entire sheet, are introduced by design with the objective of rendering the medium anisotropic. Such anisoptropy leads to acoustic directionality, which can be exploited for waveguiding or acoustic-focusing purposes.