  We report the results of a spectroscopic search for Lyman-alpha emission from
gamma-ray burst host galaxies. Based on the well-defined TOUGH sample of 69
X-ray selected Swift GRBs, we have targeted the hosts of a subsample of 20 GRBs
known from afterglow spectroscopy to be in the redshift range 1.8-4.5. We
detect Lya emission from 7 out of the 20 hosts, with the typical limiting
3sigma line flux being 8E-18 erg/cm2/s, corresponding to a Lya luminosity of
6E41 erg/s at z=3. The Lya luminosities for the 7 hosts in which we detect Lya
emission are in the range (0.6-2.3)E42 erg/s corresponding to star-formation
rates of 0.6-2.1 Msun/yr (not corrected for extinction). The rest-frame Lya
equivalent widths (EWs) for the 7 hosts are in the range 9-40A. For 6 of the 13
hosts for which Lya is not detected we place fairly strong 3sigma upper limits
on the EW (<20A), while for others the EW is either unconstrained or has a less
constraining upper limit. We find that the distribution of Lya EWs is
inconsistent with being drawn from the Lya EW distribution of bright Lyman
break galaxies at the 98.3% level, in the sense that the TOUGH hosts on average
have larger EWs than bright LBGs. We can exclude an early indication, based on
a smaller, heterogeneous sample of pre-Swift GRB hosts, that all GRB hosts are
Lya emitters. We find that the TOUGH hosts on average have lower EWs than the
pre-Swift GRB hosts, but the two samples are only inconsistent at the 92%
level. The velocity centroid of the Lya line is redshifted by 200-700 km/s with
respect to the systemic velocity, similar to what is seen for LBGs, possibly
indicating star-formation driven outflows from the host galaxies. There seems
to be a trend between the Lya EW and the optical to X-ray spectral index of the
afterglow (beta_OX), hinting that dust plays a role in the observed strength
and even presence of Lya emission. [ABRIDGED]
