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The non-trivial phase-space distribution of relic neutrinos is responsible for the erasure of primordial density perturbations on small scales, which is one of the main cosmological signatures of neutrino mass. In this paper, we present a new code, FastDF, for generating 1%-accurate particle realisations of the neutrino phase-space distribution using relativistic perturbation theory. We use the geodesic equation to derive equations of motion for massive particles moving in a weakly perturbed spacetime and integrate particles accordingly. We demonstrate how to combine geodesic-based initial conditions with the $δf$ method to minimise shot noise and clarify the definition of the neutrino momentum, finding that large errors result if the wrong parametrisation is used. Compared to standard Lagrangian methods with ad-hoc thermal motions, FastDF achieves substantial improvements in accuracy. We outline the approximation schemes used to speed up the code and to ensure symplectic integration that preserves phase-space density. Finally, we discuss implications for neutrino particles in cosmological N-body simulations. In particular, we argue that particle methods can accurately describe the neutrino distribution from $z=10^9$, when neutrinos are linear and ultra-relativistic, down to $z=0$, when they are nonlinear and non-relativistic. FastDF can be used to set up accurate initial conditions (ICs) for N-body simulations and has been integrated into the higher-order IC code monofonIC.