We discuss the dynamics of the Universe within the framework of the massive graviton cold dark matter scenario (MGCDM) in which gravitons are geometrically treated as massive particles. In this modified gravity theory, the main effect of the gravitons is to alter the density evolution of the cold dark matter component in such a way that the Universe evolves to an accelerating expanding regime, as presently observed. Tight constraints on the main cosmological parameters of the MGCDM model are derived by performing a joint likelihood analysis involving the recent supernovae type Ia data, the cosmic microwave background shift parameter, and the baryonic acoustic oscillations as traced by the Sloan Digital Sky Survey red luminous galaxies. The linear evolution of small density fluctuations is also analyzed in detail. It is found that the growth factor of the MGCDM model is slightly different (∼1–4%) from the one provided by the conventional flat ΛCDM cosmology. The growth rate of clustering predicted by MGCDM and ΛCDM models are confronted to the observations and the corresponding best fit values of the growth index (γ) are also determined. By using the expectations of realistic future x-ray and Sunyaev-Zeldovich cluster surveys we derive the dark matter halo mass function and the corresponding redshift distribution of cluster-size halos for the MGCDM model. Finally, we also show that the Hubble flow differences between the MGCDM and the ΛCDM models provide a halo redshift distribution departing significantly from the those predicted by other dark energy models. These results suggest that the MGCDM model can observationally be distinguished from ΛCDM and also from a large number of dark energy models recently proposed in the literature.