16 pages, 7 figures.-- Submitted to JCAP.-- PACS numbers: 98.65.Dx, 95.35.+d, 14.60.Pq.We have performed detailed studies of non-linear structure formation in cosmological models with light neutrinos. For the first time the effect of neutrino thermal velocities has been included in a consistent way, and the effect on the matter power spectrum is found to be significant. The effect is large enough to be measured in future, high precision surveys. Additionally, we provide a simple but accurate analytic expression for the suppression of fluctuation power due to massive neutrinos. Finally, we describe a simple and fast method for including the effect of massive neutrinos in large-scale N-body simulations which is accurate at the 1% level for Σ m_nu < 0.15 eV.We acknowledge computing resources from the Danish Center for Scientific Computing (DCSC).Peer reviewed