The propagation of cardiac electrical excitation is influenced by tissue microstructure. Reaction-diffusion computational models of cardiac electrophysiology incorporating both dynamic action potential (AP) behaviour and image-based myocardial architecture provide an approach to study the complex organisation of excitation waves within variable myocardial structures. The role of tissue microstructure (cardiomyocyte and sheetlet orientations) on organ-scale arrhythmic excitations was investigated. Five healthy rat ventricle datasets were obtained using diffusion tensor MRI (DTI). The Fenton-Karma minimal AP model was modified to reproduce the rat AP duration and restitution. Re-entrant scroll waves were initiated in the five anatomical models at ten locations for three microstructure scenarios: (i) isotropic; (ii) anisotropic; and (iii) orthotropic. Variability in anatomy and microstructure caused simulated scroll waves to self-terminate, remain tachycardia-like, or degenerate into fibrillatory activity. Whilst inclusion of DTI-based microstructure increased total scroll wave filament length to differing extents between the five hearts, overall mean filament dynamics were quantitatively similar under anisotropic and orthotropic conditions. This study highlights the important role of inter-subject structural variability.