Tunable plasmonic Dirac semimetal metamaterials featuring multiple transparency windows in the terahertz (THz) frequency region are numerically investigated in this study. The proposed multiple plasmon-induced transparency (PIT) system is composed of two types of H-shaped structures, each of which can be regarded as an independent PIT system. Unlike in the conventional metallic PIT systems, the multiple PIT effect in three-dimensional Dirac semimetals can be tuned by varying the Fermi energy instead of the geometrical parameters. The proposed multiple PIT structure could pave the way for potential applications in various areas such as multiple-frequency slow-light devices, optical sensing, and active plasmonic switching.