IP networks have become the dominant platform for video delivery. However, bandwidth-hungry video is pushing networks to their limits: costs are rising for the operators and the viewing experience is not always satisfactory for the users. When considering 3D video delivery, the previous problems are exacerbated because of the higher volume of data that must be communicated, and the difficulty in characterizing the viewing experience of the end user. Consequently, network operators may be reluctant to deliver 3D video due to costs and unclear quality improvements to their users. In this setting, the true immersive experience of 3D video remains elusive. In this paper, we focus on the efficient delivery of 3D video in terms of quality and energy cost over centrally controlled networks. As a representative example of a centrally controlled network, a software-defined network (SDN) is assumed. Our approach is based on a comprehensive network-dependent 3D quality of experience (3DQoE) model and an energy cost model for 3D video streaming. By using the developed models, we formulate the problem of energy-efficient and 3DQoE-optimized 3D video flow path routing. The particular characteristic of video/depth rate allocation presented in 3D video is embedded seamlessly into the selection of the optimal routing paths for multiple 3D video streams. The formulated problem is NP-hard and is solved with a heuristic algorithm based on the branch-and-bound method after significant reduction of the solution search space. Extensive 3D video streaming experiments are conducted over an OpenFlow-based SDN with subjective and objective evaluations and they highlight the significant benefits of the proposed approach.