We have studied the thermal conductivity κ on single crystalline samples of the antiferromagnetic monolayer cuprates R2CuO4 with R=La, Pr, Nd, Sm, Eu, and Gd. For a heat current within the CuO2 planes, i.e., for κab we find high-temperature anomalies around 250K in all samples. In contrast, the thermal conductivity κc perpendicular to the CuO2 planes, which we measured for R=La, Pr, and Gd, shows a conventional temperature dependence as expected for a purely phononic thermal conductivity. This qualitative anisotropy of κi and the anomalous temperature dependence of κab give evidence for a significant magnetic contribution κmag to the heat transport within the CuO2 planes. Our results suggest, that a large magnetic contribution to the heat current is a common feature of single-layer cuprates. We find that κmag is hardly affected by structural instabilities, whereas already weak charge carrier doping causes a strong suppression of κmag .