A novel approach to reconstructing the principal directions of a diffusion tensor field directly from magnetic resonance imaging (MRI) data using a tensor tomography data acquisition approach was developed. If tensor eigenvalues are assumed to be known, the reconstruction of principal directions requires fewer measurements than the reconstruction of the full tensor field. The tensor tomography data acquisition method (rotating diffusion gradients) leads to a unique reconstruction of principal directions, whereas the conventional MRI acquisition technique (stationary diffusion gradients) leads to an ambiguous reconstruction of principal directions when the same number of measurements are used. A computer-generated phantom was used to simulate the diffusion tensor field in the mid-ventricular region of the myocardium. The principal directions of the diffusion tensor field were assumed to align with the fibre structure of the myocardium. An iterative algorithm was used to reconstruct the principal directions. Computer simulations verify that the proposed method provides accurate reconstruction of the principal directions of a diffusion tensor field.