Is the graviton massless? This problem was addressed in the literature at a phenomenological level, using modified dispersion relations for gravitational waves, in linearized calculations around flat space. Here, we perform a detailed analysis of the gravitational waveform produced when a small particle plunges or inspirals into a large nonspinning black hole. Our results should presumably also describe the gravitational collapse to black holes and explosive events such as supernovae. In the context of a theory with massive gravitons and screening, merging objects up to 1 Gpc away or collapsing stars in the nearby galaxy may be used to constrain the mass of the graviton to be smaller than similar to 10(-23) eV, with low-frequency detectors. Our results suggest that the absence of dipolar gravitational waves from black hole binaries may be used to rule out entirely such theories.