Chromatic dispersion, which introduces pattern-dependent inter-symbol interference (ISI), appears to be a long-standing performance limiting problem in optical intensity modulation direct detection (IM/DD) transmission systems. In this paper, we propose a multiplier-free maximum likelihood sequence estimation (MLSE) equalizer for C-band double-sideband IM/DD transmission. It models the IM/DD channel with dispersion-induced ISI and Gaussian noise. A look-up table is applied to record ISI for transition probability calculation and the Viterbi algorithm for decision sequence acquisition. Specifically, to reduce the number of multipliers, a refined construction of Viterbi algorithm based on tentative path decisions is adopted, which compresses the complexity of branch metric calculation to less than 1/4 for PAM-4 format. Moreover, approximation calculation is employed to realize multiplier-free hardware implementation, which greatly reduces the hardware consumption. The proposed MLSE equalizer offers superior performance and lower complexity over conventional equalizers. In the experimental verification, we experimentally demonstrate a C-band 56-Gb/s double-sideband 4-level pulse amplitude modulation (PAM-4) IM/DD transmission over 40-km standard single mode fiber exploiting the proposed refined MLSE without any optical amplifier, filter or dispersion managed modules at the receiver end, achieving a bit-error-ratio of 2.65×10-4, which is 2.28 orders of magnitude lower than the scheme using Volterra nonlinear equalizer.