Using generalized gradient elasticity, this paper studies transverse wave characteristics of carbon nanotubes (CNTs) in a free space and embedded in a surrounding elastic medium. For flexural waves propagating in single-walled CNTs, the phase and group velocities are obtained analytically and they are scale-dependent. For single-walled CNTs surrounded by an elastic medium, waves with frequencies lower than the cut-off frequency cannot travel. For double-walled CNTs, waves of acoustic and optical modes are both present, and the cut-off frequency is dependent on the van der Waals force and the surrounding medium, but independent of the scale parameters. The phase speed of flexural waves is strongly affected by the surrounding medium for very low frequencies, and by gradient coefficients for higher frequencies. Dispersion relations are presented graphically to show the scale effects of wave speed with and without a surrounding elastic medium.