We develop the relativistic theory of hydrodynamic fluctuations for application to high-energy heavy-ion collisions. In particular, we investigate their effect on the expanding boost-invariant (Bjorken) solution of the hydrodynamic equations. We discover that correlations over a long rapidity range are induced by the propagation of the sound modes. Due to the expansion, the dispersion law for these modes is nonlinear and attenuated even in the limit of zero viscosity. As a result, there is a nondissipative wake behind the sound front which is generated by any instantaneous pointlike fluctuation. We evaluate the two-particle correlators using the initial conditions and hydrodynamic parameters relevant for heavy-ion collisions at RHIC and LHC. In principle these correlators can be used to obtain information about the viscosities because the magnitudes of the fluctuations are directly proportional to them.