The thermal motion of Si ad-dimers on Ge(001) has been studied with scanning tunneling microscopy. At room temperature the Si ad-dimers residing on top of the substrate dimer rows perform a one-dimensional random walk along the substrate dimer rows. The activation barrier for the diffusion process is estimated to be 0.83 eV. Although the preferential diffusion direction is along the substrate dimer rows, also diffusion across the rows has been observed. The latter diffusion process consists of two separate events: a jump of a Si ad-dimer from an on-top position to a position in the trough between the substrate dimer rows and a hop from a trough position to an on-top position. [S0163-1829(98)02448-5].