We study the expansion dynamics of a one-dimensional polarized Fermi gas after its sudden release from confinement using both the mean-field Bogoliubov-de Gennes and the numerically unbiased time-evolving block decimation methods. Our results show that experimentally observable spin density modulations, directly related to the presence of a Fulde-Ferrell-Larkin-Ovchinnikov state, develop during the expansion of the cloud. Our work therefore provides a robust theoretical proposal for the detection of this long-sought state.