The dynamics of water molecules monolayered on SrF2, and ZnO surfaces was investigated by quasi-elastic neutron scattering (QENS) measurements. On both SrF2, and ZnO systems, the spectra of the monolayer samples are composed of a quasi-elastic wing besides the elastic peak; those of the dried samples give only the latter peak, The line width of the quasi-elastic component of the monolayer sample for SrF2, increases with the momentum transfer Q, indicating that the observed motion comprises both rotational and translational ones. The relaxation times tau of the observed motions were compared with the literature values for water molecules monolayered on Cr2O3 and in bulk liquid water, ice, and amorphous ice that were determined by QENS and dielectric measurements. For the SrF2 and ZnO systems, the T values determined by the QENS measurements lie on the extended lines of dielectric data, while not for the Cr2O3 system. These results substantiate the validity of a structure model previously suggested, i.e., the monolayer water on Cr2O3 is a two-dimensional (2D) crystal and those on SrF2 and ZnO are 2D liquids. The T values for the SrF2 and ZnO systems are longer than those for bulk water around room temperature, and their order of magnitude crosses over at lower temperatures (amorphous ice region). This result is explained as follows. The low mobility of the monolayer water molecules around room temperature is ascribed to a decrease in activation entropy caused by limited paths for molecular motion of the monolayer water on impermeable crystal surface. The formation of the hydrogen-bond network at low temperatures is hindered by the surface field, which results in a small apparent activation energy.