We report on low-temperature heat-transport properties of the spin-1/2 triangular-lattice antiferromagnet Cs$_2$CuCl$_4$. Broad maxima in the thermal conductivity along the three principal axes, observed at about 5 K, are interpreted in terms of the Debye model, including the phonon Umklapp scattering. For thermal transport along the $b$ axis, we observed a pronounced field-dependent anomaly, close to the transition into the three-dimensional long-range-ordered state. No such anomalies were found for the transport along the $a$ and $c$ directions. We argue that this anisotropic behavior is related to an additional heat-transport channel through magnetic excitations, that can best propagate along the direction of the largest exchange interaction. Besides, peculiarities of the heat transport of Cs$_2$CuCl$_4$ in magnetic fields up to the saturation field and above are discussed.
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