We report the first measurement of the double-spin asymmetry A(LT) for charged pion electroproduction in semi-inclusive deep-inelastic electron scattering on a transversely polarized He-3 target. The kinematics focused on the valence quark region, 0.16 < x < 0.35 with 1.4 < Q(2) < 2.7 GeV2. The corresponding neutron A(LT) asymmetries were extracted from the measured He-3 asymmetries and proton over He-3 cross section ratios using the effective polarization approximation. These new data probe the transverse momentum dependent parton distribution function g(1T)(q) and therefore provide access to quark spin-orbit correlations. Our results indicate a positive azimuthal asymmetry for pi(-) production on He-3 and the neutron, while our pi(+) asymmetries are consistent with zero.