In this study, the analysis of the mechanical response of realistic fullerene-nanotube nanotruss networks with face-centered cubic geometry is performed by using molecular dynamics with reactive potentials. In particular, the mechanical properties of these novel architectures are investigated in both compressive and tensile regimes, a number of truss geometries by straining along different directions. Our atomistic simulations reveal a similar behavior under tensile stress for all the samples. Conversely, under compressive regimes the emergence of a response that depends on the orientation of load is observed together with a peculiar local instability. Due to this instability, some of these nanotruss networks present a negative Poisson ratio in compression, like re-entrant foams. Finally, the performance of these nanotruss networks is analyzed with regards to their use as impact energy absorbers, displaying properties that outperform materials traditionally used in these applications