  We show that if the gravitational Chern-Simons term couples to a massive
scalar field ($m>H$), the primordial gravitational waves (GWs) will show itself
the chirality oscillation, i.e., the amplitudes of the left- and right-handed
GWs modes will convert into each other and oscillate in their propagations.
This oscillation will eventually develop a permanent difference of the
amplitudes of both modes, which leads to nearly opposite oscillating shapes in
the power spectra of the left- and right-handed primordial GWs. We discuss its
implication to the CMB B-mode polarization.
