  We examine the propagation and flavor oscillations of neutrinos under the
influence of gravitational waves (GWs) with an arbitrary polarization. We
rederive the effective Hamiltonian for the system of three neutrino flavors
using the perturbative approach. Then, using this result, we consider the
evolution of neutrino flavors in stochastic GWs with the general energy density
spectrum.The equation for the density matrix for neutrino flavors is obtained
and solved analytically. As an application, we study the flavor content of a
neutrino beam emitted in a supernova type-II explosion. We obtain the
analytical expressions for the contributions of GWs to the neutrino fluxes and
for the damping decrement, which describes the attenuation of the fluxes to
their asymptotic values. We are able to evaluate qualitatively the effect of
various sources of stochastic GWs on the evolution of neutrino fluxes. We prove
that the major contribution is by GWs emitted by merging supermassive black
holes. The implication of the obtained results for the measurement of
astrophysical neutrinos with neutrino telescopes is discussed.
