Analogous piezoelectric networks have been shown to be effective for multimodal vibration attenuation in structures, including beams, plates, and rings. Previous studies for rings have only accounted for in-plane transverse vibration attenuation and disregarded the out-of-plane vibration modes. Furthermore, these previous numerical models and experiments have only been studied on thin rings, which ignore the effects of shear deformation and rotary inertia. As a result, these networks are not suitable for attenuating vibrations in thick rings. This study enhances on the previous electrical networks, considering both shear deformation and rotary inertia for both in-plane and out-of-plane vibrations. A new passive network topology is developed for the out-of-plane dynamics of thick rings, and the existing passive analogous network of in-plane vibration of thin rings is enhanced by considering the effects of shear deformation and rotary inertia. Combined, these new networks are capable of multimodal vibration damping of a thick ring in three-dimensions, encompassing primarily six types of vibration modes: the inextensional bending modes, the extensional modes, the thickness-shear modes, the coupled twist-bending modes, the torsional modes, and the transverse thickness-shear modes. By using piezoelectric elements to couple two separate analogous passive electrical networks derived from both the in-plane and out-of-plane governing equations of a ring and optimizing the internal resistance in each unit cell, it becomes possible to replicate the dynamics and effectively attenuate different types of vibration modes. This study serves as a theoretical foundation for implementations of passive vibration attenuation in ring structures.