A low-cost fiber Bragg grating (FBG) vibrometer specifically suited for structural monitoring and aimed at the detection of low-amplitude vibrations is presented. The optical system exploits an intensity modulation principle of operation, while signal processing techniques are used to complement the transducer to improve the performances: a recursive least-squares adaptive filter improves the noise power mitigation by 14 dB, and an efficient spectral estimator permits operating spectral analysis even under high noise conditions. With these methods, a strain sensitivity of 5.6 nε has been achieved in the ±60 µε range. Experimental assessment tests carried out in typical structural monitoring contexts have demonstrated that the developed sensor is well suited to measure mechanical perturbations of different structures.