Understanding the structure of the heart at the microscopic scale of cardiomyocytes and their aggregates provides new insights into the mechanisms of heart disease and enables the investigation of effective therapeutics. Diffusion Tensor Cardiac Magnetic Resonance (DT-CMR) is a unique non-invasive technique that can resolve the microscopic structure, organisation, and integrity of the myocardium without the need for exogenous contrast agents. However, this technique suffers from relatively low signal-to-noise ratio (SNR) and frequent signal loss due to respiratory and cardiac motion. Current DT-CMR techniques rely on acquiring and averaging multiple signal acquisitions to improve the SNR. Moreover, in order to mitigate the influence of respiratory movement, patients are required to perform many breath holds which results in prolonged acquisition durations (e.g., ~30 mins using the existing technology). In this study, we propose a novel cascaded Convolutional Neural Networks (CNN) based compressive sensing (CS) technique and explore its applicability to improve DT-CMR acquisitions. Our simulation based studies have achieved high reconstruction fidelity and good agreement between DT-CMR parameters obtained with the proposed reconstruction and fully sampled ground truth. When compared to other state-of-the-art methods, our proposed deep cascaded CNN method and its stochastic variation demonstrated significant improvements. To the best of our knowledge, this is the first study using deep CNN based CS for the DT-CMR reconstruction. In addition, with relatively straightforward modifications to the acquisition scheme, our method can easily be translated into a method for online, at-the-scanner reconstruction enabling the deployment of accelerated DT-CMR in various clinical applications.