This paper deals with the free vibration of a beam which is discretized by finite rigid granular elements and compared its exact solution to continuum local elasticity. This problem, which can be considered as a simple model to rigorously study the effect of the microstructure on the dynamic behavior of a equivalent continuum structural model, can be referred to Cosserat discrete chain or a lattice elastic model with shear interaction. This micro structured system consists of uniform grains elastically connected by shear and rotation springs. First the critical frequencies of the system which change the nature of the results, are obtained. Next, the natural frequencies of such as granular model are analytically calculated whatever the considered modes for the granular beam resting on two supports, starting from the resolution of the linear difference eigenvalue problem. It is shown that the discrete equations of this granular system, for an infinite number of grains, converge to the differential equations of the Bresse-Timoshenko beam resting on Winkler foundation (such a Bresse-Timoshenko beam can be also classified as a continuous Cosserat beam model). A gradient Bresse-Timoshenko model is constructed from continualization of the difference equations of the granular system. This continuous gradient elasticity Cosserat model is obtained from a polynomial or a rational expansion of the pseudodifferential operators, stemming from the continuualization process. The natural frequencies of the continuous gradient Cosserat models are compared with those of the discrete Cosserat model associated with the granular chain. Scale effects of the granular chain are clearly captured by the continuous gradient elasticity model. The results clarify the dependency of the beam dynamic responses to the beam length ratio.