Functional near-infrared spectroscopy (fNIRS) has attracted widespread attention as an emerging neuroimaging technology. However, the existence of motion artifacts in the fNIRS signal leads to bias in its signal processing outcomes. We proposed a tMedMor algorithm that combines the targeted median filtering (tMed) and mathematical morphology (Mor) for the removal of three motion artifacts in the fNIRS signal, namely, spike, baseline shift, and slow drift. Simulated and experimental data were used for verification, and the performance of the proposed algorithm was compared with those of several other common algorithms. Our results revealed that the tMedMor algorithm demonstrates good performance in terms of mean square error, signal-to-noise ratio, square of Pearson correlation coefficient, and peak-to-peak error, which together indicate that tMedMor can be applied as a new approach to the fNIRS signal at the preprocessing stage.