The effect of motions which are slow on the NMR time scale (in particular, lateral diffusion) on the P-31-NMR transverse relaxation behavior of POPC (1-palmitoyl-2-oleoyl-sn-glycero-3-phosphocholine) phospholipid membranes was studied by experiment and computer simulation. Two new membrane model systems (spherical supported vesicles) which exhibit some unique geometrical features and physical properties as well as the commonly used multilamellar vesicles, were applied for the measurement of the P-31-NMR anisotropic transverse relaxation times. The known geometry of the new membrane model systems under study enables a direct comparison of the experimental transverse relaxation spectra with those obtained by a spectral simulation technique which considers random diffusion on the surface of a sphere as the dominant relaxation mechanism, where the sphere size corresponds to that used in the experiments. The transverse relaxation times obtained experimentally for the three model systems studied differ drastically, where the multilamellar vesicles exhibit the longest T2 and single spherical supported bilayers show the shortest relaxation time. Both experiment and simulation indicate that lateral diffusion causes anisotropic transverse relaxation which scales with the angle theta between molecular director axis and magnetic field as I/T2 approximately Sin2theta Cos2theta. An analysis of the experimental data obtained for supported systems suggests that diffusion of the individual molecules along the surface of the sphere is superimposed by diffusion over an ''adiabatic rough'' surface (a ''quasistationary '' roughness on the time scale of the echo experiment). This provides a significant additional contribution to the transverse relaxation time T2. A comparison of single bilayers and multibilayers on a solid support, respectively, indicates that this roughness is mainly a property of the single supported bilayers, most likely due to an area mismatch between the bilayer and the surface of its solid support.