It is shown that a gravitationally bound system with a one-dimensional velocity dispersion sigma can at most dissipate a fraction ~36(sigma/c)^3 of the gravitational wave energy propagating through it. The limit is saturated for particles with a mean-free-path equal to the size of the system, such as hot protons in galaxy clusters, strongly-interacting dark matter particles in halos, or massive black holes in clusters. Even for these systems, the dissipated fraction, <10^{-6}, does not degrade the use of gravitational waves as standard sirens for cosmology. In particular, the dissipation effect is smaller by a factor ~0.03(c/sigma)^2 than the peculiar velocity uncertainty of the source within its host dynamical system. The electromagnetic counterparts that result from the dissipation are too faint to be detectable at cosmological distances.