The atomic structure of the group-III-rich surface of III-V semiconductor compounds has been under intense debate for many years, yet none of the models agrees with the experimental data available. Here we present a model for the three-dimensional structure of the (001)-c(8x2) reconstruction on InSb, InAs, and GaAs surfaces based on surface x-ray diffraction data that was analyzed by direct methods and subsequent least squares refinement. Contrary to common belief the main building blocks of the structure are not dimers on the surface but subsurface dimers in the second bilayer. This essential feature of the structure is accompanied by linear arrays of atoms on nonbulklike sites at the surface which, depending on the compounds, exhibit a certain degree of disorder. A tendency to group-III-dimer formation within these chains increases when descending the periodic table. We propose that all the c(8 x 2) reconstructions of III-V semiconductor surfaces contain the same essential building blocks.