Inelastic-neutron-scattering methods have been used to investigate the vibrational spectra of the alkali-metal atoms in M${\mathrm{C}}_{\mathrm{x}}$ compounds, where M=K, Rb, or Cs, and x=8, 24, or 36. Partial phonon densities of states have been determined, in which only in--basal-plane modes associated with the M atoms are represented. The physical origin of M-M forces is discussed. In-plane Debye-Waller factors for the intercalate atoms have been calculated from the observed spectra. For the stage-2 compounds ${\mathrm{KC}}_{24}$ and ${\mathrm{RbC}}_{24}$, and for the stage-1 compound ${\mathrm{RbC}}_{8}$, detailed measurements of the temperature dependence have been carried out, in order to study the influence of the order-disorder phase transformations which occur, respectively, at 123, 162, and 745 K. Only relatively subtle effects of the phase changes were observed, and the time scale of diffusion in the disordered phases is much longer than that of vibrations, unlike the case of real three-dimensional liquid metals.