We demonstrate that in a wide class of multilayered superconductor-ferromagnet structures (e.g., S/F, S/F/N, and S/F/F') the vanishing Meissner effect signals the appearance of the in-plane Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) modulated superconducting phase. In contrast to the bulk superconductors the FFLO instability in these systems can emerge at temperatures close to the critical one and is effectively controlled by the S layer thickness and the angle between magnetization vectors in the F/F' bilayers. The predicted FFLO state is revealed through the critical temperature oscillations vs the perpendicular magnetic field component.