In this work we address the problem of estimating 3D human pose from a single RGB image by blending a feed-forward CNN with a graphical model that couples the 3D positions of parts. The CNN populates a volumetric output space that represents the possible positions of 3D human joints, and also regresses the estimated displacements between pairs of parts. These constitute the 'unary' and 'pairwise' terms of the energy of a graphical model that resides in a 3D label space and delivers an optimal 3D pose configuration at its output. The CNN is trained on the 3D human pose dataset 3.6M, the graphical model is trained jointly with the CNN in an end-to-end manner, allowing us to exploit both the discriminative power of CNNs and the top-down information pertaining to human pose. We introduce (a) memory efficient methods for getting accurate voxel estimates for parts by blending quantization with regression (b) employ efficient structured prediction algorithms for 3D pose estimation using branch-and-bound and (c) develop a framework for qualitative and quantitative comparison of competing graphical models. We evaluate our work on the Human 3.6M dataset, demonstrating that exploiting the structure of the human pose in 3D yields systematic gains.