Continuous-wave (CW) terahertz (THz) photomixing requires compact, widely tunable, mode-hop-free driving lasers. We present a single-mode microelectromechanical system (MEMS)-tunable vertical-cavity surface-emitting laser (VCSEL) featuring an electrothermal tuning range of 64 nm (7.92 THz) that exceeds the tuning range of commercially available distributed-feedback laser (DFB) diodes (∼4.8 nm) by a factor of about 13. We first review the underlying theory and perform a systematic characterization of the MEMS-VCSEL, with particular focus on the parameters relevant for THz photomixing. These parameters include mode-hop-free CW tuning with a side-mode-suppression-ratio >50 dB, a linewidth as narrow as 46.1 MHz, and wavelength and polarization stability. We conclude with a demonstration of a CW THz photomixing setup by subjecting the MEMS-VCSEL to optical beating with a DFB diode driving commercial photomixers. The achievable THz bandwidth is limited only by the employed photomixers. Once improved photomixers become available, electrothermally actuated MEMS-VCSELs should allow for a tuning range covering almost the whole THz domain with a single system.