The flux of Galactic cosmic rays (GCRs) can undergo a Forbush decrease (FD) during the passage of a shock, sheath region, or magnetic flux rope associated with a coronal mass ejection (CME). Cosmic-ray observations during FDs can provide information complementary to in situ observations of the local plasma and magnetic field, because cosmic-ray distributions allow remote sensing of distant conditions. Here we develop techniques to determine the GCR anisotropy before and during an FD using data from neutron monitors worldwide. For example, for the FD starting on 2013 April 13, we find that at times with strong magnetic fluctuations and strong cosmic-ray scattering, there were spikes of high perpendicular anisotropy and weak parallel anisotropy. These results, along with the near constancy of parallel anisotropy across magnetic field reversals, are consistent with diffusive barriers causing the decrease in GCR flux before the arrival of the flux rope. In contrast, within the CME flux rope there was a strong parallel anisotropy in the direction predicted from a theory of drift motions into one leg of the magnetic flux rope and out the other, confirming that the anisotropy can remotely sense a large-scale flow of GCRs through a magnetic flux rope.