The mathematical modeling of damping materials based on fractional calculus has been shown to be very effective in representing the frequency dependence of the properties of these materials. In this model, the integer order derivatives in the constitutive equations of the Kelvin model are replaced by derivatives of fractional order. In this paper, we examine the response of a single degree-of-freedom system in which the damping force is proportional to a derivative of order α < 1 of the displacements. Three methods are proposed to obtain the response: the Laplace and Fourier transform methods, and an operator method that results in a series solution. Some interesting features exhibited by the oscillator’s response due to the fractional representation of the damping are unveiled.