The dynamics of a one-dimensional, highly discrete, linear array of alternating 0− and π− Josephson junctions is studied numerically, under constant bias current at zero magnetic field. The calculated current -voltage characteristics exhibit half-integer and integer zero-field-like steps for even and odd total number of junctions, respectively. Inspection of the instantaneous phases reveals that, in the former case, single π−kink excitations (discrete semi-fluxons) are supported, whose propagation in the array gives rise to the 1/2−step, while in the latter case, a pair of π−kink -π−antikink appears, whose propagation gives rise to the 1−step. When additional 2π−kinks are inserted in the array, they are subjected to fractionalization, transforming themselves into two closely spaced π−kinks. As they propagate in the array along with the single π−kink or the π−kink -π−antikink pair, they give rise to higher half-integer or integer zero-field-like steps, respectively.