To achieve this we need to find the characteristic imprints that the cosmic neutrinos leave on large scale structures in the universe. This subject was discussed by Julien Lesgourgues who knows everything about structure formation. His speech was easy and clear, even though his name could suggest otherwise. There was also a related talk by Steen Hannestad.
Neutrinos are practically collisionless. As long as they remain relativistic they free-stream. This means they tend to wipe out dark matter perturbations at scales shorter than the free- streaming length. The latter is simply related to the relic neutrino energy density. Thus, by measuring the distribution of dark matter in the universe we can find out how much neutrino component it contains.
The solid curve is the prediction of a model with cold dark matter only. The relic neutrinos would like to suppress the density fluctuations on the left-hand side of the plot. The effect would be 60% for 1eV neutrinos and 3% for 0.05 eV neutrinos. The current data are good enough to set the bound
The future is bright for dark matter. New experiments will soon provide more accurate data (Planck for the CMB, SDSS-II for galaxy surveys). Weak lensing will tell us about the time evolution of the dark matter power spectrum (see the recent article about weak lensing tomography in Nature). All this will increase the sensitivity to neutrino masses down to 0.1 eV level.
To learn more, there is a comprehensive review Julien made for Physics Reports. If you need something lighter to read on a train, take a look at this article by Steen.