We study TEM-wave propagation inside a hollow coaxial waveguide filled with a periodic composite of lossy impedance-matched right-handed (RHM) and left-handed (LHM) media. The z-direction, chosen as the direction perpendicular to the boundaries between the two media, is where the transitions and TEM-wave propagation take place. The relative permittivity epsilon(omega, z) and permeability mu(omega, z) of the periodic RHM-LHM composite vary according to an arbitrary periodic function f(z) along the z-direction. In particular, we consider two specific periodic permittivity and permeability profiles: one with abrupt RHM-LHM transitions and one with linear RHM-LHM transitions. The expected properties of impedance-matched periodic RHM-LHM structures are confirmed by the derived precise analytical solutions to Maxwell's equations, which also include solutions for the wave behavior and field components. Furthermore, a numerical study of the wave propagation over the impedance-matched graded RHM-LHM composites is performed using the COMSOL Multiphysics software. The resulting numerical simulations and analytical results were virtually identical. The present method can model smooth, realistic material transitions and includes the abrupt transition as a limiting case.