It is a well known fact that many interesting phenomena in the theory of waves in nonlinear lattices, e.g., the significant reduction of the amplitude of a propagating primary pulse or the essential growth of the phase velocity, may be explained in terms of various resonant mechanisms existing in the system (e.g. Frankel-Kontorova model). Recently, we have demonstrated analytically and numerically that similar resonant mechanisms also exist in periodically disordered granular chains with no pre-compression. Moreover, these mechanisms are responsible for the aforementioned phenomena of intensive pulse attenuation as well as speeding up of solitary waves in periodic granular chains. In our studies we have considered regular dimer chains consisting of pairs of ‘heavy’ and ‘light’ beads with no pre compression and with elastic Hertzian interaction between beads. A new family of solitary waves was discovered for these systems. These solitary waves may be considered analogous to the solitary wave of a homogeneous chain studied by Nesterenko [1], in the sense that they do not involve separations between beads, but rather satisfy special symmetries or, equivalently resonances in the dynamics. We show that these solitary waves arise from a countable infinity (we conjecture) of nonlinear anti-resonances in the dimer chains. Moreover, solitary waves in the dimers propagate faster than solitary waves in the homogeneous granular chain obtained in the limit of no mass mismatch (i.e., composed of only ‘heavy’ beads). This finding, which might seem to be counter intuitive, indicates that under certain conditions nonlinear anti-resonances can increase the speed of disturbance propagation in disordered granular media, by generating new ways of transferring energy to the far field in these media. Finally, we discuss a contrasting resonance mechanism that leads to the opposite effect, that is, very efficient shock attenuation in the dimer chain. Indeed, we show that under a certain nonlinear resonance condition a granular dimer chain can greatly reduce the amplitude of propagating pulses, through effective scattering of the energy of the pulse to higher frequencies and excitation of alternative intrinsic dynamics of the dimer. This resonance condition may be theoretically predicted and explained, and a very fair correspondence is observed between the analytical solutions and direct numerical simulations. From a practical point of view, these results can have interesting implications in applications where granular media are employed as shock transmitters or attenuators.