We present a survey for neutral atomic-carbon (C I) along gamma-ray burst (GRB) sightlines, which probes the shielded neutral gas-phase in the interstellar medium (ISM) of GRB host galaxies at high redshift. We compile a sample of 29 medium- to high-resolution GRB optical afterglow spectra spanning a redshift range through most of cosmic time from 1 <  z <  6. We find that seven (≈25%) of the GRBs entering our statistical sample have C I detected in absorption. It is evident that there is a strong excess of cold gas in GRB hosts compared to absorbers in quasar sightlines. We investigate the dust properties of the GRB C I absorbers and find that the amount of neutral carbon is positively correlated with the visual extinction, AV, and the strength of the 2175 Å dust extinction feature, Abump. GRBs with C I detected in absorption are all observed above a certain threshold of logN(H I)/cm−2 + X/H > 20.7 and a dust-phase iron column density of logN(Fe)dust/cm−2 > 16.2. In contrast to the SED-derived dust properties, the strength of the C I absorption does not correlate with the depletion-derived dust properties. This indicates that the GRB C I absorbers trace dusty systems where the dust composition is dominated by carbon-rich dust grains. The observed higher metal and dust column densities of the GRB C I absorbers compared to H2- and C I-bearing quasar absorbers is mainly a consequence of how the two absorber populations are selected, but is also required in the presence of intense UV radiation fields in actively star-forming galaxies.