Josephson junctions containing three ferromagnetic layers with noncollinear magnetizations between adjacent layers carry spin-triplet supercurrent under certain conditions. The signature of the spin-triplet supercurrent is a relatively slow decay of the maximum supercurrent as a function of the thickness of the middle ferromagnetic layer. In this work we focus on junctions where the middle magnetic layer is a [Co/Pd](N) multilayer with perpendicular magnetic anisotropy (PMA), while the outer two layers have in-plane anisotropy. We compare junctions where the middle PMA layer is or is not configured as a synthetic antiferromagnet (PMA-SAF). We find that the supercurrent decays much more rapidly with increasing the number N of [Co/Pd] bilayers in the PMA-SAF junctions compared to the PMA junctions. Similar behavior is observed in junctions containing [Co/Ni](N) PMA multilayers. We model that behavior by assuming that each Co/Pd or Co/Ni interface acts as a partial spin filter, so that the spin-triplet supercurrent in the PMA junctions becomes more strongly spin-polarized as N increases, while the supercurrent in the PMA-SAF junctions is suppressed with increasing N. We also address a question raised in a previous work regarding how much spin-singlet supercurrent is transmitted through our nominally spin-triplet junctions. We do that by comparing spin-triplet junctions with similar junctions where the order of the magnetic layers has been shuffled. The results of this work are expected to be helpful in designing spin-triplet Josephson junctions for use in cryogenic memory.