Osteoporosis induced fractures occur worldwide about every 3 seconds. Vertebral compression fractures are early signs of the disease and considered risk predictors for secondary osteoporotic fractures. We present a detection method to opportunistically screen spine-containing CT images for the presence of these vertebral fractures. Inspired by radiology practice, existing methods are based on 2D and 2.5 D features but we present, to the best of our knowledge, the first method for detecting vertebral fractures in CT using automatically learned 3D feature maps. The presented method explicitly localizes these fractures allowing radiologists to interpret its results. We train a voxel-classification 3D Convolutional Neural Network (CNN) with a training database of 90 cases that has been semi-automatically generated using radiologist readings that are readily available in clinical practice. Our 3D method produces an Area Under the Curve (AUC) of 95% for patient-level fracture detection and an AUC of 93% for vertebra-level fracture detection in a five-fold cross-validation experiment.