The (3He,t) charge-exchange reaction populating Jπ=2− states has been examined at 420 MeV incident energy for a series of double-β decaying nuclei, i.e., 76Ge, 82Se, 96Zr, 100Mo, 128Te, 130Te, and 136Xe. The measurements were carried out at the Grand Raiden spectrometer of the Research Center for Nuclear Physics at the University Osaka with typical spectral resolution of 30–40 keV. It is found that the charge-exchange reaction leading to 2− spin-dipole states is selective to the στ part of the interaction much similar to the observed selectivity to Gamow-Teller transitions. In the present case, the ΔL=1 peak cross sections at finite momentum transfers are used to extract the spin-isospin part of the low-lying transition strength near the Fermi surface (i.e., Ex≤5 MeV). Relative strength values are confronted with various model calculations, i.e., the interacting shell model, the quasiparticle random-phase approximation, and the Fermi surface quasiparticle model. The impact on the nuclear matrix elements for the neutrinoless double-β decay is discussed.