In this work, we develop a multipath-based simultaneous localization and mapping (SLAM) method that can directly be applied to received radio signals. In existing multipath-based SLAM approaches, a channel estimator is used as a preprocessing stage that reduces data flow and computational complexity by extracting features related to multipath components (MPCs). We aim to avoid any preprocessing stage that may lead to a loss of relevant information. The presented method relies on a new statistical model for the data generation process of the received radio signal that can be represented by a factor graph. This factor graph is the starting point for the development of an efficient belief propagation (BP) method for multipath-based SLAM that directly uses received radio signals as measurements. Simulation results in a realistic scenario with a single-input single-output (SISO) channel demonstrate that the proposed direct method for radio-based SLAM outperforms state-of-the-art methods that rely on a channel estimator.