We demonstrate continuous tuning of a terahertz quantum cascade wire laser over a broad range of $\sim 330\mathrm{GHz}$ ( $\sim 8.6\%$ of the $3.85\mathrm{THz}$ center frequency) with single-mode operation. Tuning is achieved with a narrow laser ridge ( $\sim {\lambda }_o/8$ ) and a moveable MEMS (microelectromechanical systems) side object (plunger) to manipulate the transverse optical mode. The frictionless MEMS plunger enables continuous and reversible tuning. We demonstrate $\sim 30\mathrm{GHz}$ redshift tuning using a silicon plunger and $\sim 300\mathrm{GHz}$ blueshift tuning using a metal plunger. The blueshift tuning range is limited by the bandwidth of the gain medium rather than the tuning mechanism.