Heavy smokers undergoing screening with low-dose chest CT are affected by cardiovascular disease as much as by lung cancer. Low-dose chest CT scans acquired in screening enable quantification of atherosclerotic calcifications and thus enable identification of subjects at increased cardiovascular risk. This paper presents a method for automatic detection of coronary artery, thoracic aorta and cardiac valve calcifications in low-dose chest CT using two consecutive convolutional neural networks. The first network identifies and labels potential calcifications according to their anatomical location and the second network identifies true calcifications among the detected candidates. This method was trained and evaluated on a set of 1744 CT scans from the National Lung Screening Trial. To determine whether any reconstruction or only images reconstructed with soft tissue filters can be used for calcification detection, we evaluated the method on soft and medium/sharp filter reconstructions separately. On soft filter reconstructions, the method achieved F1 scores of 0.89, 0.89, 0.67, and 0.55 for coronary artery, thoracic aorta, aortic valve and mitral valve calcifications, respectively. On sharp filter reconstructions, the F1 scores were 0.84, 0.81, 0.64, and 0.66, respectively. Linearly weighted kappa coefficients for risk category assignment based on per subject coronary artery calcium were 0.91 and 0.90 for soft and sharp filter reconstructions, respectively. These results demonstrate that the presented method enables reliable automatic cardiovascular risk assessment in all low-dose chest CT scans acquired for lung cancer screening.