Fractional telegrapher's equation is reinterpreted in the setting of heat conduction phenomena and reobtained by considering the energy balance equation and fractional Cattaneo heat conduction law, generalized by taking into account the history of temperature gradient as well. Using the Laplace transform method, fractional telegrapher's equation is solved on semi-bounded domain for the zero initial condition and solution is obtained as a convolution of forcing temperature on the boundary and impulse response. Some features of such obtained solution are examined.