We propose a fully automated algorithm based on a deep-learning framework enabling screening of a Coronary Computed Tomography Angiography (CCTA) examination for confident detection of the presence or complete absence of atherosclerotic plaque of the coronary arteries. The system starts with extracting the coronary arteries and their branches from CCTA datasets and representing them with multi-planar reformatted volumes; pre-processing and augmentation techniques are then applied to increase the robustness and generalization ability of the system. A 3-Dimensional Convolutional Neural Network (3D-CNN) is utilized to model pathological changes (e.g., calcification) in coronary arteries/branches. The system then learns the discriminatory features between vessels with and without atherosclerosis. The discriminative features at the final convolutional layer are visualized with a saliency map approach to localize the visual clues related to atherosclerosis. We have evaluated the system on a reference dataset representing 247 patients with atherosclerosis and 246 patients free of atherosclerosis. With 5-fold cross-validation, an accuracy = 90.9%, with Positive Predictive Value = 58.8%, Sensitivity = 68.9%, Specificity of 93.6%, and Negative Predictive Value = 96.1% are achieved at the artery/branch level with a threshold of 0.5. The average area under the curve = 0.91. The system indicates a high negative predictive value, which may be potentially useful for assisting physicians in identifying patients with no coronary atherosclerosis that need no further diagnostic evaluation.