  We compute the screening length for quarkonium mesons moving through an
anisotropic, strongly coupled N=4 super Yang-Mills plasma by means of its
gravity dual. We present the results for arbitrary velocities and orientations
of the mesons, as well as for arbitrary values of the anisotropy. The
anisotropic screening length can be larger or smaller than the isotropic one,
and this depends on whether the comparison is made at equal temperatures or at
equal entropy densities. For generic motion we find that: (i) mesons dissociate
above a certain critical value of the anisotropy, even at zero temperature;
(ii) there is a limiting velocity for mesons in the plasma, even at zero
temperature; (iii) in the ultra-relativistic limit the screening length scales
as $(1-v^2)^\epsilon$ with \epsilon =1/2, in contrast with the isotropic result
\epsilon =1/4.
