This paper addresses two key limitations of the unscented Kalman filter (UKF) when applied to the simultaneous localization and mapping (SLAM) problem: The cubic, in the number of states, computational complexity, and the inconsistency of the state estimates. In particular, we introduce a new sampling strategy that minimizes the linearization error and whose computational complexity is constant (i.e., independent of the size of the state vector). As a result, the overall computational complexity of UKF-based SLAM becomes of the same order as that of the extended Kalman filter (EKF) when applied to SLAM. Furthermore, we investigate the observability properties of the linear-regression-based model employed by the UKF, and propose a new algorithm, termed the Observability-Constrained (OC)-UKF, that improves the consistency of the state estimates. The superior performance of the OC-UKF compared to the standard UKF and its robustness to large linearization errors are validated by extensive simulations.