Methodologies to acquire three-dimensional velocity fields are becoming increasingly available, generating large datasets of steady state and transient flows of engineering and/or biomedical interest. This paper presents a novel linear filter for three-dimensional velocity acquisitions, which eliminates the spurious velocity divergence due to measurement errors. The noise reduction properties of the associated linear operator are discussed together with the treatment of boundary conditions. Examples show the application of the technique to real velocity fields acquired through volumetric Particle Image Velocimetry. The effectiveness of the filter is demonstrated by application to synthetic velocity fields obtained from analytical solutions and computations. The filter eliminates about half of the noise, without artificial smoothing of the original data, and conserves localized flow features.