We introduce a new discrete-ordinate scheme for solving the general relativistic (GR) Boltzmann transport equation in the context of core-collapse supernovae (CCSNe). Our algorithm avoids the need to spell out the complicated advection terms in energy and angle that arise when the transport equation is formulated in spherical polar coordinates, in the comoving frame, or in a GR space-time. We instead approach the problem by calculating the advection of neutrinos across momentum space using an intuitive particle-like approach that has excellent conservation properties and fully accounts for Lorentz boosts, GR effects, and grid geometry terms. In order to avoid the need for a global implicit solution, time integration is performed using a locally implicit Lax-Wendroff scheme that correctly reproduces the diffusion limit. This will facilitate the use of our method on massively parallel distributed-memory architectures. We have verified the accuracy and stability of our scheme with a suite of test problems in spherical symmetry and axisymmetry. To demonstrate that the new algorithm works stably in CCSN simulations, we have coupled it to the GR hydrodynamics code coconut and present a first demonstration run of a [Formula: see text] progenitor with a reduced set of neutrino opacities. © 2020 The Author(s) Published by Oxford University Press on behalf of the Royal Astronomical Society.