Do electromagnetic waves always propagate along null geodesics?

We find exact solutions to Maxwell equations written in terms of four-vector potentials in non-rotating, as well as in Gödel and Kerr spacetimes. We show that Maxwell equations can be reduced to two uncoupled second-order differential equations for combinations of the components of the four-vector potential. Exact electromagnetic waves solutions are written on given gravitational field backgrounds where they evolve. We find that in non-rotating spherical symmetric spacetimes, electromagnetic waves travel along null geodesics. However, electromagnetic waves on Gödel and Kerr spacetimes do not exhibit that behavior.