We consider S>3/2 kagome and triangular lattice magnets with strong easy-axis single-ion anisotropy D and antiferromagnetic exchange J. When D>>J, the low energy states selected by the anisotropy map onto configurations of the corresponding classical Ising antiferromagnet. Subleading O(J;{3}S/D;{2}) multispin interaction arising from the transverse quantum dynamics makes the low-temperature behavior very different from the well-known classical case: The kagome magnet goes into a semiclassical spin-liquid state with distinctive and unusual short-range correlations below a crossover temperature T;{*} approximately 0.08J;{3}S/D;{2}, while the triangular magnet undergoes a first-order transition at T_{c} approximately 0.1J;{3}S/D;{2} to an orientationally ordered collinear state that gives rise to a novel zero-magnetization plateau for small magnetic fields along the easy axis. Possible experimental implications are also discussed.