The forward Euler's integration method is widely used to compute the trajectories of the control variables in conventional predictive current control (PCC) techniques. However, the computational error caused by the forward Euler's method proportionally increases with the sampling time. Hence, the current tracking ability of conventional PCC techniques deteriorates when operating with a large sampling time. Furthermore, the conventional PCC methods produce more switching transitions to reduce the current tracking error, thereby leading to a high switching frequency operation. To overcome these problems, Heun's integration method is proposed for the PCC in this article. The proposed method has predictor and corrector stages to minimize the computational error caused by the large sampling time during the prediction process of the control variables. Thereby, the proposed Heun's method-based PCC techniques produce fewer switching transitions to minimize the current tracking error. Consequently, this results in a decrease in switching frequency. The proposed PCC method is applied to a four-level multilevel inverter (4L-MLI). The discrete-time system models are developed using Heun's integration method to implement the proposed PCC. The working philosophy of the proposed method is demonstrated through the dSPACE/DS1103 controlled 4L-MLI laboratory prototype. The experimental performance of the proposed PCC is compared with that of the conventional PCC in terms of average switching frequency, current tracking error, current harmonic distortion, and transient response time.