  It has been shown that a cosmological background with an anisotropic stress
tensor, appropriate for a free streaming thermal neutrino background, can damp
primordial gravitational waves after they enter the horizon, and can thus
affect the CMB B-mode polarization signature due to such tensor modes. Here we
generalize this result, and examine the sensitivity of this effect to non-zero
neutrino masses, extra neutrino species, and also a possible relativistic
background of axions from axion strings. In particular, additional neutrinos
with cosmologically interesting neutrino masses at the O(1) eV level will
noticeably reduce damping compared to massless neutrinos for gravitational wave
modes with $k\tau_0 \approx 100-200$, where $\tau_0 \approx 2/H_0$ and $H_0$ is
the present Hubble parameter, while an axion background would produce a
phase-dependent damping distinct from that produced by neutrinos.
