Dispersion of a nonelectrolyte solute due to the electroosmotic flow in long straight microchannels was analyzed theoretically. A version of the Aris-Taylor procedure was employed to predict the dispersion coefficient for arbitrary geometry of the microchannel cross section. The analysis was conducted using a thin double-layer approximation, which is valid when the Debye length is much smaller than the characteristic dimensions of the cross section. For thin double layers, the obtained results describe the electroosmotic dispersion for arbitrary surface potential, electrolyte type, and cross-section geometry. Dispersion for several cases of the cross-section geometries was discussed. It was shown that, for given values of the surface potential and the Debye length, both the cross-section geometry and the electrolyte content of the driven solution substantially affect the dispersion of a nonelectrolyte solute. In the relevant particular cases, the obtained results agree with predictions of the previous theories.