The influence of the atomic-level structure of electrode surfaces on electrochemical oscillations has been studied in a system of H2O2 reduction on Pt electrodes in acidic solutions. A current oscillation of another type, named oscillation E, has been found to appear for an atomically flat single-crystal Pt(111) electrode, in addition to previously reported oscillations, named oscillations A and B. Oscillation E does not appear for atomically flat Pt(100), Pt(110), polycrystalline Pt, and Pt(111) with atomically nonflat surfaces. Mathematical simulation by use of a model including an autocatalytic effect of adsorbed OH for dissociative adsorption of H2O2, as a possible explanation, has reproduced the appearance of oscillation E, as well as observed correlations between the appearance of oscillation E and the magnitudes of H2O2-reduction current and "negative" resistance. It is discussed that an efficient autocatalytic mechanism works at the atomically flat Pt(111) surface, which is responsible for the appearance of oscillation E at this surface.