We present the kinematics of a sample of 398 DA white dwarfs from the SPY project (ESO SN Ia Progenitor surveY) and discuss kinematic criteria for distinguishing of thin-disk, thick-disk, and halo populations. This is the largest homogeneous sample of white dwarfs for which 3D space motions have been determined. Since the percentage of old stars among white dwarfs is higher than among main-sequence stars, they are presumably valuable tools in studies of old populations, such as the halo and the thick disk. Studies of white-dwarf kinematics can help to determine the fraction of the total mass of our Galaxy contained in the form of thick-disk and halo white dwarfs, an issue which is still under discussion. Radial velocities and spectroscopic distances obtained by the SPY project were combined with our measurements of proper motions to derive 3D space motions. Galactic orbits and further kinematic parameters were computed. We calculated individual errors of kinematic parameters by means of a Monte Carlo error propagation code. Our kinematic criteria for assigning population membership were deduced from a sample of F and G stars taken from the literature, for which chemical criteria can be used to distinguish between a thin-disk, a thick-disk and a halo star. Our kinematic population classification scheme is based on the position in the U-V-velocity diagram, the position in the J_z_-eccentricity diagram, and the Galactic orbit. We combined this with age information and found seven halo and 23 thick-disk white dwarfs in this brightness limited sample. Another four rather cool white dwarfs probably also belong to the thick disk. Correspondingly 2% of the white dwarfs belong to the halo and 7% to the thick disk. The mass contribution of the thick-disk white dwarfs is found to be substantial, but is insufficient to account for the missing dark matter.