In physical systems characterized by approximate conservation laws, transport properties are strongly dominated by slow decay rates of the nearly conserved currents. I will discuss a particular manifestation of this phenomenon in the low-temperature heat transport of one-dimensional spin-1/2 chains coupled to phonons. The contribution of spin excitations (spinons) to the thermal conductivity is analyzed, and found to be in remarkable agreement with prominent features of the experimental data in SrCuO compounds. I will also discuss predictions of dramatic effects induced by an externally applied magnetic field in such systems.