In this paper, a new genre of spin-transfer torque (STT) MRAM is proposed, in which bidirectional writing is achieved using thermoelectrically controlled magnonic current as an alternative to conventional electric current. The device uses a magnetic tunnel junction (MTJ), which is adjacent to a nonmagnetic metallic and a ferrite film. This film stack is heated or cooled by a Peltier element, which creates a bidirectional magnonic pulse in the ferrite film. Conversion of magnons to spin current occurs at the ferrite-metal interface, and the resulting STT is used to achieve fast ( ~ nanosecond) precessional switching of the ferromagnetic free layer in the MTJ. Compared to the electric-current-driven STT-MRAM with perpendicular magnetic anisotropy (PMA), thermoelectric STT-MRAM reduces the overall magnetization switching energy by more than 40% for nanosecond switching, combined with a write error rate (WER) of less than 10-9 and a lifetime of ten years or higher. The combination of higher thermal activation energy, subnanosecond read/write speed, improved tunneling magnetoresistance (TMR), and tunnel barrier reliability make thermoelectric STT-MRAM a promising choice for future nonvolatile memory applications.