  We explore a recently proposed locally resonant granular system bearing
harmonic internal resonators in a chain of beads interacting via Hertzian
elastic contacts. In this system, we propose the existence of two types of
configurations: (a) small-amplitude periodic traveling waves and (b)
dark-breather solutions, i.e., exponentially localized, time periodic states
mounted on top of a non-vanishing background. We also identify conditions under
which the system admits long-lived bright breather solutions. Our results are
obtained by means of an asymptotic reduction to a suitably modified version of
the so-called discrete p-Schr{\"o}dinger (DpS) equation, which is established
as controllably approximating the solutions of the original system for large
but finite times (under suitable assumptions on the solution amplitude and the
resonator mass). The findings are also corroborated by detailed numerical
computations. A remarkable feature distinguishing our results from other
settings where dark breathers are observed is the complete absence of
precompression in the system, i.e., the absence of a linear spectral band.
