Determining bulk residual elastic strains is a topic of critical importance for predictive modelling and materials testing. Although there are numerous approaches to obtaining this information the process is often time-consuming and frequentlyinvolves destructive sectioning of the sample. This paper presents the latest developments in Bragg-edge neutron strain tomography which offers a rapid, non-destructive means of determining residual elastic strains in polycrystalline materials with high spatial resolution.Neutron strain tomography utilises the fact that the energy-resolved transmission spectrum of thermal neutrons froma polycrystalline sample displays well-defined, sudden jumps in intensity known as 'Bragg-edges', which reveal information about the average residual elastic strain throughout the sample volume. By measuring the spatially resolved transmission spectrum for a number of different sample orientations it is possible to recover the underlying three-dimensional residual elastic strain profile. This process is analogous to conventional absorption tomography where a three-dimensional map of the sample density is recovered from a lower order two-dimensional set of integral absorption measurements. Using the newly developed microchannel plate TimePixneutron detectorin time-of-flight experiments, elastic strains can in principle be recovered with 10 um spatial resolution. The current work demonstrates a model-free method for direct inversion of the average strain profiles obtained from Bragg-edge measurements using results from linear elasticity theory and the Radon transform, which is the basis for absorption tomography reconstructions. We demonstrate this method in simulation on an axisymmetric sample with analytic strain profile before applying it to a 'real-world' sample where the results are validated against conventional neutron diffraction and hole drilling measurements.