Large contributions of the thermal conductivity due to spins, κspin, in low-dimensional spin systems are expected to be utilized as highly thermal-conducting materials. Single crystals of one-dimensional spin systems RbFeCl3 with ferromagnetic (FM) chains and CsFeBr3 with antiferromagnetic (AF) chains have been prepared in order to observe the contribution of κspin to the thermal conductivity. The thermal conductivity parallel to spin chains along the c-axis, κ//c, of RbFeCl3 has been found to be enhanced around 3 K and 10 K by the application of magnetic field. In the thermal conductivity perpendicular to the c-axis, κ⊥c, of RbFeCl3, on the other hand, only one peak around 3 K has been found to be enhanced by the application of magnetic field. Since κ⊥c is mainly owing to the thermal conductivity due to phonons, κphonon, it has been concluded that the peak of κ//c around 10 K in magnetic fields is due to the contribution of κspin. For CsFeBr3, it has been found that κ//c shows two peaks around 3 K and 25 K while κ⊥c shows one peak around 12 K in zero field. This indicates that there is a marked contribution of κspin to κ//c. However, the details are not yet clear. It has been found that κ//c in RbFeCl3 is enhanced by the application of magnetic field while κ//c in CsFeBr3 is suppressed. The difference may be due to the difference between FM chains in RbFeCl3 and AF chains in CsFeBr3.