After an upgrade in 2019-2021, the LHCb experiment is taking data in Run 3 (2022-2026) with an instantaneous luminosity of proton-proton collisions of 2×1033 cm2s-1. This article presents the Radiation Monitoring System (RMS-R3) for controlling the beam and background conditions at LHCb. It runs continuously during the detector's operation, and independently of the main LHCb data acquisition. Its design is based on robust and radiation-hard Metal Foil Detector technology. The RMS-R3 monitors the instantaneous luminosity and its evolution. The analysis of the RMS-R3 Run 3 data demonstrates its linear response with a high reproducibility in a five orders of magnitude dynamic range of luminosity over a long period of operation.