Most recent approaches to monocular 3D pose estimation rely on Deep Learning. They either train a Convolutional Neural Network to directly regress from an image to a 3D pose, which ignores the dependencies between human joints, or model these dependencies via a max-margin structured learning framework, which involves a high computational cost at inference time. In this paper, we introduce a Deep Learning regression architecture for structured prediction of 3D human pose from monocular images or 2D joint location heatmaps that relies on an overcomplete autoencoder to learn a high-dimensional latent pose representation and accounts for joint dependencies. We further propose an efficient Long Short-Term Memory network to enforce temporal consistency on 3D pose predictions. We demonstrate that our approach achieves state-of-the-art performance both in terms of structure preservation and prediction accuracy on standard 3D human pose estimation benchmarks.