Spin-polarized currents play a key role in spintronics. Recently, it has been found that antiferromagnets with a non-spin-degenerate band structure can efficiently spin-polarize electric currents, even though their net magnetization is zero. Among the antiferromagnetic metals with magnetic space group symmetry supporting this functionality, the noncollinear antiferromagnetic antiperovskites ANMn$_3$ (A = Ga, Ni, Sn, and Pt) are especially promising. This is due to their high Neel temperatures and a good lattice match to perovskite oxide substrates, offering possibilities of high structural quality heterostructures based on these materials. Here, we investigate the spin polarization of antiferromagnetic ANMn$_3$ metals using first-principles density functional theory calculations. We find that the spin polarization of the longitudinal currents in these materials is comparable to that in widely used ferromagnetic metals, and thus can be exploited in magnetic tunnel junctions and spin transfer torque devices. Moreover, for certain film growth directions, the out-of-plane transverse spin currents with a giant charge-to-spin conversion efficiency can be achieved, implying that the ANMn$_3$ antiperovskites can be used as efficient spin sources. These properties make ANMn$_3$ compounds promising for application in spintronics.