The dynamic characteristic analysis model of damped carbon nanotubes on viscoelastic foundations was built by using Euler-Bernoulli beams. The nonlocal viscoelastic theories, general Maxwell viscoelastic model, velocity-dependent external damping model and viscoelastic foundation model were employed to deduce the governing equation of Euler-Bernoulli beams for dynamic characteristics analysis of carbon nanotubes. On the basis of Kelvin-Voigt model, new general analytical expressions for the natural frequencies of damped carbon nanotubes with no foundation and full foundation were obtained respectively and some typical special cases at full foundation were discussed. Then a transfer function method was developed to obtain a closed-form and uniform solution for the vibration governing equation under arbitrary boundary conditions. Considering a single-walled carbon nanotube as a numerical example, the first four natural frequencies with different boundary conditions were obtained, and the effects of the nonlocal and viscoelastic constants, the foundation stiffness coefficient and length on the natural frequencies and damping factors were analyzed. Results demonstrate the efficiency of the proposed model and the analysis methods in solving dynamic problems of damped carbon nanotubes on viscoelastic foundations.