When strained Stranski-Krastanow islands are used as “self-assembled quantum dots,” a key goal is to control the island position. Here we show that nanoscale grooves can control the nucleation of epitaxial Ge islands on Si(001), and can drive lateral motion of existing islands onto the grooves, even when the grooves are very narrow and shallow compared to the islands. A position centered on the groove minimizes energy. We use as prototype grooves the trenches which form naturally around islands. During coarsening, the shrinking islands move laterally to sit directly astride that trench. In subsequent growth, we demonstrate that islands nucleate on the “empty trenches” which remain on the surface after complete dissolution of the original islands.