Coronary artery centerline extraction in cardiac CT angiography (CCTA) images is a prerequisite for evaluation of stenoses and atherosclerotic plaque. In this work, we propose an algorithm that extracts coronary artery centerlines in CCTA using a convolutional neural network (CNN). In the proposed method, a 3D dilated CNN is trained to predict the most likely direction and radius of an artery at any given point in a CCTA image based on a local image patch. Starting from a single seed point placed manually or automatically anywhere in a coronary artery, a tracker follows the vessel centerline in two directions using the predictions of the CNN. Tracking is terminated when no direction can be identified with high certainty. The CNN is trained using manually annotated centerlines in training images. No image preprocessing is required, so that the process is guided solely by the local image values around the tracker's location. The CNN was trained using a training set consisting of 8 CCTA images with a total of 32 manually annotated centerlines provided in the MICCAI 2008 Coronary Artery Tracking Challenge (CAT08). Evaluation was performed within the CAT08 challenge using a test set consisting of 24 CCTA test images in which 96 centerlines were extracted. The extracted centerlines had an average overlap of 93.7% with manually annotated reference centerlines. Extracted centerline points were highly accurate, with an average distance of 0.21 mm to reference centerline points. Based on these results the method ranks third among 25 publicly evaluated methods in CAT08. In a second test set consisting of 50 CCTA scans acquired at our institution (UMCU), an expert placed 5448 markers in the coronary arteries, along with radius measurements. Each marker was used as a seed point to extract a single centerline, which was compared to the other markers placed by the expert. This showed strong correspondence between extracted centerlines and manually placed markers. In a third test set containing 36 CCTA scans from the MICCAI 2014 Challenge on Automatic Coronary Calcium Scoring (orCaScore), fully automatic seeding and centerline extraction was evaluated using a segment-wise analysis. This showed that the algorithm is able to fully-automatically extract on average 92% of clinically relevant coronary artery segments. Finally, the limits of agreement between reference and automatic artery radius measurements were found to be below the size of one voxel in both the CAT08 dataset and the UMCU dataset. Extraction of a centerline based on a single seed point required on average 0.4 ± 0.1 s and fully automatic coronary tree extraction required around 20 s. The proposed method is able to accurately and efficiently determine the direction and radius of coronary arteries based on information derived directly from the image data. The method can be trained with limited training data, and once trained allows fast automatic or interactive extraction of coronary artery trees from CCTA images.