An elementary method, based on the use of complex variables, is proposed for solving the equation of motion of a simple harmonic oscillator. The method is first applied to the equation of motion for an undamped oscillator and it is then extended to the more important case of a damped oscillator. It is finally shown that the method can readily be extended to cover the case of the forced damped oscillator as well. The proposed method of approach presents many advantages, which should make it appealing for classroom discussions. For one, it is easy to implement and solve, particularly when there is no damping force; this feature is very important for an initial rigorous discussion of this important system. The results are easy to obtain without difficulty by students familiar with only the basic properties of complex numbers and of the exponential function. It is claimed that the approach has pedagogical merit for an early-level college or undergraduate presentation.