Preterm infants' limb-pose estimation is a crucial but challenging task, which may improve patients' care and facilitate clinicians in infant's movements monitoring. Work in the literature either provides approaches to whole-body segmentation and tracking, which, however, has poor clinical value, or retrieve a posteriori limb pose from limb segmentation, increasing computational costs and introducing inaccuracy sources. In this paper, we address the problem of limb-pose estimation under a different point of view. We proposed a 2D fully-convolutional neural network for roughly detecting limb joints and joint connections, followed by a regression convolutional neural network for accurate joint and joint-connection position estimation. Joints from the same limb are then connected with a maximum bipartite matching approach. Our analysis does not require any prior modeling of infants' body structure, neither any manual interventions. For developing and testing the proposed approach, we built a dataset of four videos (video length = 90 s) recorded with a depth sensor in a neonatal intensive care unit (NICU) during the actual clinical practice, achieving median root mean square distance [pixels] of 10.790 (right arm), 10.542 (left arm), 8.294 (right leg), 11.270 (left leg) with respect to the groundtruth limb pose. The idea of estimating limb pose directly from depth images may represent a future paradigm for addressing the problem of preterm-infants' movement monitoring and offer all possible support to clinicians in NICUs.