Using the hard-core boson approach, we study the pump-probe spectrum of molecular assemblies carrying Frenkel excitons of arbitrary structure and dimension. We present a rigorous justification of the hard-core boson approach by using the Agranovich-Toshich transformation from paulions to bosons. The resulting two-exciton Green function is used to derive a general expression of the assembly's pump-probe spectrum. We show that this expression considerably simplifies for ordered systems occupying a lattice, where we allow for the occurrence of more than one equivalent molecule in the unit cell (Davydov components). Explicit semianalytical expressions are given for the pump-probe spectrum of linear chains with alternating dipoles, ring aggregates, chains with a herringbone structure, and monolayers. In the analysis of these expressions, we focus on the overall shape of the spectrum and on the effects of probe polarization. It is shown that relaxation during the pump-probe delay time may drastically affect the pump-probe spectrum.