Using inelastic scattering of light (Bragg spectroscopy), we study the low-energy excitations of strongly correlated phases of ultracold bosons on the cross-over from correlated 1D superfluids to Mott insulators. As it is commonly performed in solid-state physics, the use of such a probe allows us to extract important information about the atomic many-body state. In particular we show that we can extract information about the dynamical structure factor S(q,ω) and about the one-particle spectral function A(q,ω) from the Bragg spectra. This technique could be extended to study more exotic correlated phases of ultracold atoms.