An {iterative weighted reliability two-step majority logic decoding} (IWRTS-MLGD) algorithm for two-step majority-logic (TS-MLG)-decodable cyclic codes is presented. In contrast to other message passing decoding algorithms that utilize real number operations, our proposed decoding algorithm requires only logical operations and integer additions. Therefore, large computational complexities can be reduced. For moderate-length TS-MLG-decodable cyclic codes, the proposed algorithm aided with soft information and a scaling factor outperforms the hard-decision TS-MLGD algorithm and hard-decision BCH codes with similar length by 1.2- and 1.0-dB, respectively.