The Reconfigurable Intelligent Surface (RIS) constitutes one of the prominent technologies for the 6th Generation (6G) of wireless communications. It is envisioned to enhance signal coverage in cases where obstacles block direct communication from Base Stations (BSs), and when high carrier frequencies are used that are sensitive to attenuation losses. In the literature, the exploitation of RISs based on traditional coherent demodulation, relies on the availability of accurate Channel State Information (CSI). Given that CSI estimation, a multi-antenna BS or a dedicated orchestration controller jointly computes the pre-coder/combiner and the RIS configuration. The latter tasks require a significant amount of time and resources, which may not be affordable when the channel is time-varying or the CSI is not accurate enough. In this paper, we consider the uplink between a single-antenna user and a multi-antenna BS, and present a novel RIS-empowered Orthogonal Frequency Division Multiplexing (OFDM) communication system, which is based on differential phase shift keying combined with random phase configurations at the RIS, thus, avoiding the channel estimation and any complex optimization processes. This feature renders our RIS-enabled system operation proposal suitable for high noise and/or mobility scenarios. Considering both an idealistic and a realistic channel model, analytical expressions for the Signal-to-Interference and Noise Ratio (SINR) and the Symbol Error Probability (SEP) of the proposed non-coherent RIS-empowered communication system are presented. Our extensive computer simulation results verify the accuracy of the presented analysis and showcase the proposed system`s performance superiority over coherent demodulation in different mobility and spatial correlation scenarios.