This is the fourth of a series of papers in which we derive simultaneous constraints on cosmological parameters and X-ray scaling relations using observations of the growth of massive, X-ray flux-selected galaxy clusters. Our data set consists of 238 clusters drawn from the ROSAT All-Sky Survey and incorporates extensive follow-up observations using the Chandra X-ray Observatory. Here we examine the constraints on neutrino properties that are enabled by the precise and robust constraint on the amplitude of the matter power spectrum at low redshift available from our data. In combination with cluster gas mass fraction, cosmic microwave background, supernova and baryon acoustic oscillation data, and incorporating conservative allowances for systematic uncertainties, we limit the species-summed neutrino mass, Mν, to < 0.33 eV at 95.4 per cent confidence in a spatially flat, cosmological constant (ΛCDM) model. In a flat ΛCDM model where the effective number of neutrino species, Neff, is allowed to vary, we find Neff= 3.4+0.6−0.5 (68.3 per cent confidence, incorporating a direct constraint on the Hubble parameter from Cepheid and supernova data). We also obtain results with additional degrees of freedom in the cosmological model, in the form of global spatial curvature (Ωk) and a primordial spectrum of tensor perturbations (r and nt). The results are not immune to these generalizations; however, in the most general case we consider, in which Mν, Neff, curvature and tensors are all free, we still obtain Mν < 0.70 eV and Neff= 3.7 ± 0.7 (at, respectively, the same confidence levels as above). These results agree well with recent work using independent data and highlight the importance of measuring cosmic structure and expansion at low as well as high (z∼ 1100) redshifts. Although our cluster data extend to redshift z= 0.5, the direct effect of neutrino mass on the growth of structure at late times is not yet detected at a significant level.