Accurate knowledge of the thermal conductivity (k) of biological tissues is important for cryopreservation, thermal ablation, and cryosurgery. Here, we adapt the 3ω method - widely used for rigid, inorganic solids - as a reusable sensor to measure k of soft biological samples two orders of magnitude thinner than conventional tissue characterization methods. Analytical and numerical studies quantify the error of the commonly used "boundary mismatch approximation" of the bi-directional 3ω geometry, confirm that the generalized slope method is exact in the low-frequency limit, and bound its error for finite frequencies. The bi-directional 3ω measurement device is validated using control experiments to within ±2% (liquid water, standard deviation) and ±5% (ice). Measurements of mouse liver cover a temperature ranging from -69°C to +33°C. The liver results are independent of sample thicknesses from 3 mm down to 100 μm and agree with available literature for non-mouse liver to within the measurement scatter.