By means of known constitutive relations for the D and H fields, which include both electric quadrupole and magnetic dipole terms, an eigenvalue theory is developed for electromagnetic wave propagation in an absorbing chiral medium. The theory allows the polarization eigenvectors to be determined, which the medium supports, and also their refractive indices and absorption coefficients. Uniaxial and cubic crystals, as well as an ideal gas, are treated in this way, and expressions are derived for their circular birefringence and dichroism in terms of property tensors of the medium. These expressions are shown to be independent of the arbitrary origin to which the multipole moments are referred. The theory is also applicable to artificial chiral materials which are "optically active" in the microwave regime.