We present the atomic structure of the c(8 x 2) reconstructions of InSb-, InAs-, and GaAs-(001) surfaces as determined by surface x-ray diffraction using direct methods. Contrary to common belief, group III dimers are not prominent on the surface, instead subsurface dimerization of group III atoms takes place in the second bilayer, accompanied by a major rearrangement of the surface atoms above the dimers to form linear arrays. By varying the occupancies of four surface sites the (001)-c(8 x 2) reconstructions of III-V semiconductors can be described in a unified model.