  Using the Newman-Penrose formalism, we obtain the explicit expressions for the polarization modes of weak, plane gravitational waves with a massive graviton. Our analysis is restricted for a specific bimetric theory whose term of mass, for the graviton, appears as an effective extra contribution to the stress-energy tensor. We obtain for such kind of theory that the extra states of polarization have amplitude several orders of magnitude smaller than the polarizations purely general relativity (GR), h_(+) and h_(x), in the VIRGO-LIGO frequency band. This result appears using the best limit to the graviton mass inferred from solar system observations and if we consider that all the components of the metric perturbation have the same amplitude h. However, if we consider low frequency gravitational waves (e.g., f_(GW) ~ 10^(-7) Hz), the extra polarization states produce similar Newman-Penrose amplitudes that the polarization states purely GR. This particular characteristic of the bimetric theory studied here could be used, for example, to directly impose limits on the mass of the graviton from future experiments that study the cosmic microwave background (CMB). 
Polarization states of gravitational waves with a massive graviton