Bevel-tip steerable needles are a promising new technology for improving accuracy and accessibility in minimally invasive medical procedures. As yet, 3D needle steering has not been demonstrated in the presence of tissue deformation and uncertainty, despite the application of progressively more sophisticated planning algorithms. This paper presents a feedback controller that steers a needle along 3D helical paths, and varies the helix radius to correct for perturbations. It achieves high accuracy for targets sufficiently far from the needle insertion point; this is counterintuitive because the system is highly underactuated and not locally controllable. The controller uses a model predictive control framework that chooses a needle twist rate such that the predicted helical trajectory minimizes the distance to the target. Fast branch and bound techniques enable execution at kilohertz rates on a 2GHz PC. We evaluate the controller under a variety of simulated perturbations, including imaging noise, needle deflections, and curvature estimation errors. We also test the controller in a 3D finite element simulator that incorporates deformation in the tissue as well as the needle. In deformable tissue examples, the controller reduced targeting error by up to 88% compared to open-loop execution.