In this paper, the programmable signal propagation paradigm, enabled by Reconfigurable Intelligent Surfaces (RISs), is exploited for high accuracy $3$-Dimensional (3D) user localization with a single multi-antenna base station. Capitalizing on the tunable reflection capability of passive RISs, we present a two-stage user localization method leveraging the multi-reflection wireless environment. In the first stage, we deploy an off-grid compressive sensing approach, which is based on the atomic norm minimization, for estimating the angles of arrival associated with each RIS, which is followed, in the second stage, by a maximum likelihood location estimation initialized with a least-squares line intersection technique. The presented numerical results showcase the high accuracy of the proposed 3D localization method, verifying our theoretical Cramér Rao lower bound analysis.