We describe the operational principle of and experimentally demonstrate a narrowband, wavelength-tunable polarization monitor based on a fiber acousto-optic tunable filter. Two orthogonally vibrating acoustic waves are used to create a variable polarizer that can be used to measure the state of polarization of an incident narrowband light source. The accuracy of the polarization monitor is measured at two different wavelengths in reference to a commercial polarimeter and is shown to be within 5%.