A proton nuclear magnetic resonance (NMR) relaxation study of molecular dynamics in the liquid crystal 4-octylphenyl 2-chloro-4-(4-cyanobenzoyloxy)benzoate (DB8Cl) is presented. DB8Cl molecules possess a strong polar terminal group and form, in addition to the nematic phase, three different smectic phases: bilayer smectic A, bilayer smectic C, and anticliniclike smectic C phase. The proton spin-lattice relaxation times were measured in all mesophases over a broad frequency range of six decades by applying conventional and fast field-cycling NMR techniques. The parameters obtained in the analysis of the experimental data give quantitative information on molecular motions, particularly for the tilted smectic phases of DB8Cl. In contrast to former conjectures, we found that the low-frequency relaxation in the bilayer smectic C phases results from director fluctuations about the layer normal, which occur without distortion of the layers, and from layer undulations, similar to those in the smectic A phase. In the low-temperature bilayer smectic C phase, a considerable slowing-down of molecular translational diffusion is observed. It confirms indirectly the anticlinic character of this mesophase. Measurements of angular dependence of the relaxation times at 60 MHz support the conclusions obtained from the frequency dispersion data.