Motivated by the recent breakthrough of the detection of Gravitational Waves (GW) from coalescent black holes by the aLIGO interferometers, we study the propagation of GW in the D-material universe, which we have recently shown to be compatible with large-scale structure and inflationary phenomenology. The medium of D-particles induces an effective mass for the graviton, as a consequence of the formation of recoil-velocity field condensates due to the underlying Born-Infeld dynamics. There is a competing effect, due to a super-luminal refractive index, as a result of the gravitational energy of D-particles acting as a dark-matter component, with which propagating gravitons interact. We examine conditions for the condensate under which the latter effect is sub-leading. We argue that if quantum fluctuations of the recoil velocity are relatively strong, which can happen in the current era of the universe, then the condensate, and hence the induced mass of the graviton, can be several orders of magnitude larger than the magnitude of the cosmological constant today. Hence, we constrain the graviton mass using aLIGO and pulsar-timing observations (which give the most stringent bounds at present). In such a sub-luminal graviton case, there is also a gravitational Cherenkov effect for ordinary high-energy cosmic matter, which is further constrained by means of ultra-high-energy cosmic ray observations. Assuming cosmic rays of extragalactic origin, the bounds on the quantum condensate strength, based on the gravitational Cherenkov effect, are of the same order as those from aLIGO measurements, in contrast to the case where a galactic origin of the cosmic rays is assumed, in which case the corresponding bounds are much weaker.