The deployment of reconfigurable intelligent surfaces (RISs) in a communication system provides control over the propagation environment, which facilitates the augmentation of a multitude of communication objectives. As these performance gains are highly dependent on the applied phase shifts at the RIS, accurate channel state information at the transceivers is imperative. However, not only do RISs traditionally lack signal processing capabilities, but their end-to-end channels also consist of multiple components. Hence, conventional channel estimation (CE) algorithms become incompatible with RIS-aided communication systems as they fail to provide the necessary information about the channel components, which are essential for a beneficial RIS configuration. To enable the full potential of RISs, we propose to use tensor-decomposition-based CE, which facilitates smart configuration of the RIS by providing the required channel components. We use canonical polyadic (CP) decomposition, that exploits a structured time domain pilot sequence. Compared to other state-of-the-art decomposition methods, the proposed Semi-Algebraic CP decomposition via Simultaneous Matrix Diagonalization (SECSI) algorithm is more time efficient as it does not require an iterative process. The benefits of SECSI for RIS-aided networks are validated with numerical results, which show the improved individual and end-to-end CE accuracy of SECSI.