The vibrational properties of a face-centered cubic granular crystal of monodisperse particles are predicted using a discrete model as well as two micropolar models, first the classical Cosserat and second an enhanced Cosserat-type model, that properly takes into account all degrees of freedom at the contacts between the particles. The continuum models are derived from the discrete model via a micro-macro transition of the discrete relative displacements and particle rotations to the respective continuum field variables. Next, only the long wavelength approximations of the models are compared and, considering the discrete model as reference, the Cosserat model shows inconsistent predictions of the bulk wave dispersion relations. This can be explained by an insufficient modeling of sliding mode of particle interactions in the Cosserat model. An enhanced micropolar model is proposed including only one new elastic tensor from the more complete second order gradient micropolar theory. This enhanced micropolar model then involves the minimum number of elastic constants to consistently predict the dispersion relations in the long wavelength limit.