Controlling the ion velocity in an ion sheath by applying an alternating current (AC) voltage to an electrode and/or a substrate is critical in plasma material processes. To externally control the velocity distribution of incident ions on a substrate, the application of tailored-waveform AC voltages instead of sinusoidal voltages has garnered interest in recent years. In this study, to investigate temporal changes in ion-velocity distributions, we developed a time-resolved laser-induced fluorescence spectroscopy (LIF) system using a continuous-wave diode laser as an excitation-laser source. A time-resolved LIF system entails the capture of temporally continuous and spectrally discrete LIF spectra during an AC voltage cycle. By measuring temporal changes in the LIF signal intensity at various excitation-laser wavelengths, the argon-ion velocity distribution near the electrode following the AC voltage can be characterized. The results of applying sinusoidal, triangular, and rectangular bias waveforms indicate that the LIF measurement scheme proposed herein can be used to investigate the dynamic behavior of ion-velocity distributions controlled by tailored-waveform AC voltages.