  Motivated by recent experiments on a compound {displaying Ising-like
short-range correlations on the triangular lattice, we study the anisotropic
easy-axis spin-$1/2$ Heisenberg model on the triangular and kagome lattice} by
performing numerical calculations of finite-temperature properties, in
particular of static spin structure factor and of thermodynamic quantities, on
systems with up to 36 sites. On the triangular lattice, the low-temperature
spin structure factor {exhibits long-range} spin correlations in the whole
range of anisotropies, whereas thermodynamic quantities reveal a crossover upon
increasing the anisotropy, most pronounced in the vanishing generalized Wilson
ratio in the easy-axis regime. In contrast, on the kagome lattice, the spin
structure factor is short-range, and thermodynamic quantities evolve steadily
between the easy-axis and the isotropic case, consistent with the
interpretation in terms of {a} spin liquid.
