  We study Dirac neutrinos propagating in rotating background matter. First we
derive the Dirac equation for a single massive neutrino in the noninertial
frame, where matter is at rest. This equation is written in the effective
curved space-time corresponding to the corotating frame. We find the exact
solution of the Dirac equation. The neutrino energy levels for
ultrarelativistic particles are obtained. Then we discuss several neutrino mass
eigenstates, with a nonzero mixing between them, interacting with rotating
background matter. We derive the effective Schr\"odinger equation governing
neutrino flavor oscillations in rotating matter. The new resonance condition
for neutrino oscillations is obtained. We also examine the correction to the
resonance condition caused by the matter rotation.
