  Context: Before the launch of the Swift satellite, the majority of the
gamma-ray burst (GRB) afterglows for which Ly-alpha was redshifted into the
observable spectrum showed evidence for a damped Ly-alpha absorber. This small
sample indicated that GRBs explode either in galaxies, or regions within them,
having high neutral hydrogen column densities.
  Aims: To increase the spectroscopic sample of GRBs with z > 2 and hence
establish the N(HI) distribution along GRB lines-of-sight.
  Methods: We have obtained six z > 2 GRB afterglow spectra and fitted the
Ly-alpha absorption line in each case to determine N(HI). This has been
complemented with 12 other Swift N(HI) values from the literature.
  Results: We show that the peak of the GRB N(HI) distribution is qualitatively
consistent with a model where GRBs originate in Galactic-like molecular clouds.
However, a systematic difference, in particular an excess of low column-density
systems compared to the predictions, indicates that selection effects and
conditions within the cloud (e.g. strong ionization) influence the observed
N(HI) range. We also report the discovery of Ly-alpha emission from the GRB
060714 host, corresponding to a star-formation rate of approximately 0.8
M_Sun/year. Finally, we present accurate redshifts of the six bursts: z = 3.240
+/- 0.001 (GRB 050319), z = 2.198 +/- 0.002 (GRB 050922C), z = 3.221 +/- 0.001
(GRB 060526), z = 3.425 +/- 0.002 (GRB 060707), z = 2.711 +/- 0.001 (GRB
060714) and z = 3.686 +/- 0.002 (GRB 060906).
