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We study the role of chiral nuclear forces on the propagation of neutrinos in hot neutron matter. In particular, we analyze the convergence of the dynamical structure factor and the neutrino mean free path with the order of the power counting of the chiral forces, as well as the role of the regulator cut-off of these forces in the determination of these quantities. Single-particle energies and chemical potentials needed to calculate the dynamical structure factor are obtained within the Brueckner--Hartree--Fock approximation extended to finite temperature. Our results show that the dynamical structure factor and the neutrino mean free path depend on the cut-off only when the chiral potential is considered at leading order (LO) and next-to leading order (NLO), becoming this dependence strongly reduced at higher orders in the chiral power counting due to the role of three-nucleon forces that start to contribute at next-to-next-to leading order (N$^2$LO) being, in particular, almost negligible at next-to-next-to-next-to leading order (N$^3$LO). The neutrino mean free path is found to converge up to densities slightly below $\sim 0.15$ fm$^{-3}$ when increasing the order of the chiral power counting, although no signal of convergence is found for densities above this value. The uncertainty associated with our order-by-order nuclear many-body calculation of the neutrino mean free path is roughly estimated from the difference between the results obtained at N$^2$LO and N$^3$LO, finding that it varies from about a few centimeters at low densities up to a bit less than $2$ meters at the largest one considered in this work, $0.3$ fm$^{-3}$.