Monoclinic thulium-doped magnesium monotungstate, Tm3+:MgWO4, is promising for efficient power-scalable continuous-wave (CW) and passively Q-switched lasers at >2 μm. Under diode pumping at 802 nm, a compact CW laser based on Z-cut Tm:MgWO4 generated 3.09 W at 2022-2034 nm with a slope efficiency of 50% which represents the highest output power ever achieved with this type of laser host. Stable passive Q-switching of the Tm:MgWO4 laser is demonstrated for the first time, to the best of our knowledge, using single-walled carbon nanotubes, graphene, and Cr2+:ZnS saturable absorbers. Using the latter, the best performance is achieved with 16.1 μJ/13.6 ns pulses at 2017.8 nm with a maximum average output power of 0.87 W and a peak power of 1.18 kW.