Reconfigurable intelligent surfaces (RISs) have emerged as a promising solution that can provide dynamic control over the propagation of electromagnetic waves. The RIS technology is envisioned as a key enabler of sixth-generation networks by offering the ability to adaptively manipulate signal propagation through the smart configuration of its phase shift coefficients, thereby optimizing signal strength, coverage, and capacity. However, the realization of this technology's full potential hinges on the accurate acquisition of channel state information (CSI). In this paper, we propose an efficient CSI prediction framework for a RIS-assisted communication system based on the machine learning (ML) transformer architecture. Architectural modifications are introduced to the vanilla transformer for multivariate time series forecasting to achieve high prediction accuracy. The predicted channel coefficients are then used to optimize the RIS phase shifts. Simulation results present a comprehensive analysis of key performance metrics, including data rate and outage probability. Our results confirm the effectiveness of the proposed ML approach and demonstrate its superiority over other baseline ML-based CSI prediction schemes such as conventional deep neural networks and long short-term memory architectures, albeit at the cost of slightly increased complexity.