CsMnBr3 is a quasi-one-dimensional Heisenberg antiferromagnet. We present results of the magnetic inelastic response of CsMnBr3 across the magnetic Brillouin zone by neutron scattering techniques. Well-defined spin-wave modes, characteristic of one-dimensional magnetic insulators, are found over most of the zone in the paramagnetic phase at 15 K. The zone centre response is not sharp, but peaks in the scattering function S(k, ω) are found. No excitation branch going to zero energy at zero wavevector is found. At small wavevectors there is a mode with energy 1.7 ± 0.2 meV and we use it to identify planar anisotropy in this system. The mid-zone response as a function of temperature is analyzed in terms of a generalized Langevin equation approach (Mori formulation) to the dynamics of the one-dimensional Heisenberg antiferromagnet. The theory contains no adjustable parameters. Our results are compared with two truncation schemes of the theory. We report qualitative agreement with the theories, including the observation of upwards renormalization of spin-wave energy with temperature. Quantitative agreement is less than good for either truncation scheme.