This paper examines the oscillatory behavior of complex viscoelastic systems with power law like relaxation behavior. Specifically, we use the fractional Maxwell model, consisting of a spring and fractional dashpot in series, which produces a power-law creep behavior and a relaxation law following the Mittag-Leffler function. The fractional dashpot is characterized by a parameter $\beta$, continuously moving from the pure viscous behavior when $\beta$= 1 to the purely elastic response when $\beta$ = 0. In this work, we study the general response function and focus on the oscillatory behavior of a fractional Maxwell system in four regimes: Stress impulse, strain impulse, step stress, and driven oscillations. The solutions are presented in a format analogous to the classical oscillator, showing how the fractional nature of relaxation changes the long-time equilibrium behavior and the short-time transient solutions. We specifically test the critical damping conditions in the fractional regime, since these have a particular relevance in biomechanics.