A well-characterized Na-vermiculite clay, containing zero, one, or two molecular layers of water between the clay platelets, has been studied by quasielastic neutron scattering (QENS). Experiments were carried out at a temperature of 300 K in two different scattering geometries; the clay platelets being at 45 and 135 degrees angles to the incident beam in order to make the elastic Q-vector perpendicular and parallel, respectively, to the clay platelets for a scattering angle of 90 degrees (Q approximate to 1.33 Angstrom(-1)). The resulting QENS spectra show that almost no hydrogen motion occurs perpendicular to the clay platelets on the experimental time scale (about 2-40 ps). The two-H2O layer vermiculite exhibits a planar rotational motion of water molecules, forming hydration shells around the Na ions, and a basically two-dimensional translational jump-diffusion motion. The translational motion was modeled using the Gaussian jump-length distribution model, resulting in a mean jump length of 1.1 Angstrom and an average residence time of 2.3 ps. Using these values we obtain an effective diffusion coefficient of 8.8* 10(-10) m(2)/s, which is only a factor 2-3 lower than for bulk water. The correlation time of the rotational motion was estimated to approximately 27 ps. In the case of the one-H2O layer vermiculite we were only able to observe a planar rotational motion with a rotational correlation time of 16 ps, i.e., faster than in the two-H2O layer vermiculite. This suggests that a smaller number of water molecules are involved in the rotational process in the one-H2O layer vermiculite, and furthermore that the translational motion, if existent, is too slow to be observed on the experimental time scale.