Here we describe and provide diffusion magnetic resonance imaging (dMRI) data that was acquired in neural tissue and a physical phantom. Data acquired in biological tissue includes: fixed rat brain (acquired at 9.4 T) and spinal cord (acquired at 16.4 T) and in normal human brain (acquired at 3 T). This data was recently used for evaluation of diffusion kurtosis imaging (DKI) contrasts and for comparison to diffusion tensor imaging (DTI) parameter contrast. The data has also been used to optimize b-values for ex vivo and in vivo fast kurtosis imaging. The remaining data was obtained in a physical phantom with three orthogonal fiber orientations (fresh asparagus stems) for exploration of the kurtosis fractional anisotropy. However, the data may have broader interest and, collectively, may form the basis for image contrast exploration and simulations based on a wide range of dMRI analysis strategies.