Transient temperature profiles for long rods of lossy dielectric materials with thermally-dependent dielectric properties exposed to uniform plane waves are obtained. Maxwell's equation and the heat equation are simultaneously solved using the finite element method to predict the power absorbed and the resulting temperature rise in samples of square and circular cross-section. Following the method introduced recently, we derive an exact radiation boundary condition which is independent of the rod cross-section. For a cylindrical sample, the boundary condition is imposed on the cylinder itself. For a square rod, the boundary condition is imposed on a cylinder containing the rod. The temperature dependence of dielectric properties and sample dimensions appreciably influence heating patterns. For square samples, the edges focus radiation, causing preferential heating at the edges. This effect is pronounced for larger samples. In addition, the incident wave polarization influences the heating of the rod. For waves where the electric field is polarized along the long axis of the sample (TMz polarization) the power absorbed is higher than when the electric field is perpendicular to the axis (TEz polarization). A case involving runaway heating is also investigated.