Magnetorheological elastomers (MREs) are polymers reinforced by ferromagnetic particles that show magnetic dependent behavior. Mixing MREs with reinforcing fibers can create a new class of material so-called “MRE composites, MRECs” with additional functionalities and properties. Here, using a Generalized Maxwell model, we proposed a new magnetic-dependent rheological model by considering the hysteresis phenomenon for MREs to predict the dynamic damping responses of MREC plates reinforced by fibers in the frequency domain. We also investigated the influence of magnetic flux intensity, the volume fraction of the fiber, the orientation angle of the fibers, the number of layers, as well as the fiber-to-matrix stiffness ratio on the natural frequency, loss factor, and mode shapes of MRECs plates. Our results suggest that homogenously increasing the elastic properties of the MRECs through the spatial distribution of fibers and changing the fiber-to-matrix stiffness ratio can effectively tailor the dynamic properties of MRECs. Tailoring these properties can provide additional freedom for the fabrication of 4D-printed MRE-based composites.