 This study investigates the occurrence of anomalous diffusive transport associated with acoustic wave fields propagating through highly scattering periodic media and shows that it can occur in perfectly periodic media and in the absence of disorder when the dispersion behavior is characterized by an isotropic, partial, band gaps. The analysis of the fundamental physical mechanism leading to this unexpected behavior is performed via a combination of deterministic, stochastic, and fractional order models and results indicate that the anomalous transport can occur in selected frequency ranges when the acoustic waves not only become diffusive but its intensity distribution acquires a distinctive Levi-alpha, stable profile having pronounced heavy tails. The behavior is well captured by a fractional diffusive transport model whose order can be obtained by the analysis of the heavy tails. This article was authored by Salvatore Mihi Sen and Fabio Sempalotti.