By means of a stress imposed rheometer coupled with a "vibrating cell," generating a Brownian motion at a macroscopic scale into the samples, we have shown that dense-phase vibrated powders exhibit rheological behaviors archetypal of non-Newtonian viscoelastic fluids. These behaviors have been accurately described through a free volume structural model based on simple "stick-slip" granular interactions. As a result, the evolution of the steady-state viscosity has been accurately expressed as a function of the shear rate, the frictional stress, the granular pressure, the mass of the samples, the vibration frequency, the vibration energy, the intergranular contact network mean life, and the free volume distribution. The model is consistent with Hookean, Coulombian, and Newtonian limits and is not only descriptive but also explicative and predictive of the encountered phenomena. In particular, a "time-granular temperature superposition principle," theoretically predicted by the model, has been experimentally verified, the "granular temperature" being controlled through the vibration energy and frequency. Moreover, this superposition principle has been precisely described by a "Vogel-Fulcher-Tammann" law, leading to very close analogies with molecular systems near their glass transition point.