It has been suggested that one may build an X-ray polarimeter by exploiting the polarization dependence of the angular distribution of L-shell fluorescence photons. In this paper the sensitivity of this approach to polarimetry is examined theoretically. The calculations are applied to several detection schemes using imaging proportional counters that would have direct application in X-ray astronomy. It is found, however, that the sensitivity of this method for measuring X-ray polarization is too low to be of use for other than laboratory applications.