In this study, the vibration and damping analysis of a three-layered sandwich plate with composite face layers and a viscoelastic core is considered. The governing equations and related boundary conditions are derived in terms of sectional force and moment resultants by using the principle of virtual work for free vibrations of the plate. The eigenvalue problem defined by these equations is solved by using the generalized differential quadrature method to obtain the frequencies and loss factors. Results are compared with the ones that exist in the literature for three plate problems. Lastly, a parametric analysis is carried out for sandwich plates with carbon fiber reinforced plastic face layers and a frequency-dependent viscoelastic core. Viscoelastic behavior is modeled based on the five-parameter fractional Zener model. The master curves and material properties of four different viscoelastic polymers are obtained from the experimental data that exist in the literature. The effects of these polymeric damping materials on the vibration and damping characteristics of the sandwich plate are thoroughly studied. The eigenproblem is also solved by using the finite element method to verify the results of the generalized differential quadrature method.