We report on low-temperature heat-transport properties of the spin-1/2 triangular-lattice antiferromagnet Cs2CuCl4. 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 found 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. In addition, peculiarities of the heat transport of Cs2CuCl4 in magnetic fields up to the saturation field and above are discussed.