We report the controlled arrangement of self-organized Ge dots on patterned Si(001) substrates. Selective epitaxial growth (SEG) of Si was first carried out in a molecular-beam epitaxy system with a Si2H6 gas source and a Ge Knudsen cell. Si SEG mesas were formed in [110]-oriented Si windows followed by subsequent Ge growth. The atomic force microscopic (AFM) results showed that Ge dots with a regular spacing were perfectly aligned on the ridges of Si stripe mesas. Only one row of dots was seen on each ridge of a stripe mesa with a mesa base width ranging from 0.5 to 0.9 mu m. The formation of regularly spaced Ge dots may be attributed to the balance between the strain energy of the dots and the repulsive interaction of the neighboring dots. Using preferential nucleation scheme, we demonstrate the placement of the dots at the predetermined sites. We also discuss the mechanism of the self-registration of the self-organized Ge dots.