Magnetic tunnel junctions (MTJs) are a key technology in modern spintronics because they are the basis of read-heads of modern hard disk drives, nonvolatile magnetic random access memories, and sensor applications. In this paper, we demonstrate that tunneling magnetoresistance can influence terahertz (THz) wave propagation through a MTJ. In particular, various magnetic configurations between parallel state and antiparallel state of the magnetizations of the ferromagnetic layers in the MTJ have the effect of changing the conductivity, making a functional modulation of the propagating THz electromagnetic fields. Operating in the THz frequency range, a maximal modulation depth of 60% is reached for the parallel state of the MTJ with a thickness of 77.45 nm, using a magnetic field as low as 30 mT. The THz conductivity spectrum of the MTJ is governed by spin-dependent electron tunneling. It is anticipated that the MTJ device and its tunability scheme will have many potential applications in THz magnetic modulators, filtering, and sensing.