Nanophononic materials have recently arisen as a promising way for controlling heat transport, mirroring the results in macroscopic phononic materials for sound transmission, filtering and attenuation applications. Here we present a Finite Element numerical simulation of the transient propagation of an acoustic Wave-Packet in a 2D nanophononic material, which allows to identify the effect of the nanostructuration on the acoustic attenuation length and thus on the transport regime for the vibrational energy. Assuming elastic behavior in the matrix and in the inclusions, we find that the rigidity contrast between them not only tunes the apparent attenuation length of the wave packet along its main trajectory, but gives rise to different behaviours, from weak to strong scattering, and waves pinning. As a consequence, different energy transport regimes can be identified in the three-parameter space of the excitation frequency, inclusions size and rigidity contrast, leading to the identification of a combination of parameters allowing for the shortest attenuation distance. These results could have applications both in the field of acoustic insulation, and for the control of heat transfer.