Cardiac diffusion tensor imaging (cDTI) is highly prone to image corruption, yet robust-fitting methods are rarely used. Single voxel outlier detection (SVOD) can overlook corruptions that are visually obvious, perhaps causing reluctance to replace whole-image shot-rejection (SR) despite its own deficiencies. SVOD's deficiencies may be relatively unimportant: corrupted signals that are not statistical outliers may not be detrimental. Multiple voxel outlier detection (MVOD), using a local myocardial neighbourhood, may overcome the shared deficiencies of SR and SVOD for cDTI while keeping the benefits of both. Here, robust fitting methods using M-estimators are derived for both non-linear least squares and weighted least squares fitting, and outlier detection is applied using (i) SVOD; and (ii) SVOD and MVOD. These methods, along with non-robust fitting with/without SR, are applied to cDTI datasets from healthy volunteers and hypertrophic cardiomyopathy patients. Robust fitting methods produce larger group differences with more statistical significance for MD, FA, and E2A, versus non-robust methods, with MVOD giving the largest group differences for MD and FA. Visual analysis demonstrates the superiority of robust-fitting methods over SR, especially when it is difficult to partition the images into good and bad sets. Synthetic experiments confirm that MVOD gives lower root-mean-square-error than SVOD.