This experimental study aimed to characterize the thermal properties of ex vivo porcine and bovine kidney tissues in steady-state heat transfer conditions in a wider thermal interval (23.2–92.8 °C) compared to previous investigations limited to 45 °C. Thermal properties, namely thermal conductivity (k) and thermal diffusivity (α), were measured in a temperature-controlled environment using a dual-needle probe connected to a commercial thermal property analyzer, using the transient hot-wire technique. The estimation of measurement uncertainty was performed along with the assessment of regression models describing the trend of measured quantities as a function of temperature to be used in simulations involving heat transfer in kidney tissue. A direct comparison of the thermal properties of the same tissue from two different species, i.e., porcine and bovine kidney tissues, with the same experimental transient hot-wire technique, was conducted to provide indications on the possible inter-species variabilities of k and α at different selected temperatures. Exponential fitting curves were selected to interpolate the measured values for both porcine and bovine kidney tissues, as well as for both k and α. The results show that the k and α values of the tissues remained rather constant from room temperature up to the onset of water evaporation, and a more marked increase was observed afterward. Indeed, at the highest investigated temperatures, i.e., 90.0–92.8 °C, the average k values were subject to 1.2- and 1.3-fold increases, compared to their nominal values at room temperature, in porcine and bovine kidney tissue, respectively. Moreover, at 90.0–92.8 °C, 1.4- and 1.2-fold increases in the average values of α, compared to baseline values, were observed for porcine and bovine kidney tissue, respectively. No statistically significant differences were found between the thermal properties of porcine and bovine kidney tissues at the same selected tissue temperatures despite their anatomical and structural differences. The provided quantitative values and best-fit regression models can be used to enhance the accuracy of the prediction capability of numerical models of thermal therapies. Furthermore, this study may provide insights into the refinement of protocols for the realization of tissue-mimicking phantoms and the choice of tissue models for bioheat transfer studies in experimental laboratories.