We report on the theoretical investigation of the plasmonic wave propagation in the coaxial cylindrical cables fabricated of both right-handed medium [with $ϵ>0$ , $\mu >0$ ] and left-handed medium [with $ϵ\left(\omega \right)<0$ , $\mu \left(\omega \right)<0$ ], using a Green’s-function (or a response function) theory in the absence of an applied magnetic field. The Green’s-function theory generalized to be applicable to such quasi-one-dimensional systems enables us to derive explicit expressions for the corresponding response functions (associated with the electromagnetic fields), which can in turn be used to derive various physical properties of the system. The confined plasmonic wave excitations in such multi-interface structures are characterized by the electromagnetic fields that are localized at and decay exponentially away from the interfaces. A rigorous analytical diagnosis of the general results in diverse situations leads us to reproduce exactly the previously well-known results in other geometries, obtained within the different theoretical frameworks. As an application, we present several illustrative examples on the dispersion characteristics of the confined (and extended) plasmonic waves in single- and double-interface structures made up of dispersive metamaterials interlaced with conventional dielectrics. These dispersive modes are also substantiated through the computation of local as well as total density of states. It is observed that the dispersive components enable the system to support the simultaneous existence of s- and p-polarization modes in the system. Such effects as this one are solely attributed to the negative-index metamaterials and are otherwise impossible. The readers will also notice the explicit μ-dependence of the dispersion relations for the s-polarization modes, obtained under special limits in some cases, for the single- and double-interface systems. The elegance of the theory lies in the fact that it does not require the matching of the boundary conditions and in its simplicity and the compact form of the desired (analytical) results.