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In recent years, thermal wave propagation in periodic structures and thermal metamaterials has attracted significant attention because of their potential for targeted control and redirection of thermal energy. However, the classical Fourier model is inadequate for transient thermal phenomena and wave-like behavior, particularly at microscale dimensions, because it assumes an infinite heat propagation speed. This study investigates dynamic heat transfer in periodic metamaterial structures using the classical Fourier law and two non-Fourier formulations: the Cattaneo–Vernotte (CV) and Dual-Phase-Lag (DPL) models. The main novelty is the application of the DPL model to thermal bandgap analysis while accounting for phase lags between heat flux and temperature gradient associated with phonon interactions. Dispersion curves across the first Brillouin zone and frequency response analyses revealed a complete thermal bandgap from 2531.6 to 5229.2 Hz, accompanied by pronounced wave attenuation. Numerical simulations performed independently in COMSOL Multiphysics and MATLAB showed excellent agreement, confirming the reliability of the results. Transient analyses over 0.2–9 s and phase-lag ratios of 0.001–1 demonstrated that increasing the ratio shifts the bandgap toward higher frequencies and enhances diffusive behavior relative to wave-like propagation. The results demonstrate that deliberate topological rearrangement of void geometries within a single-material domain can substantially widen the forbidden band and effectively block thermal waves, without requiring complex multi-material compositions. The proposed design offers a promising route for thermal management devices, wave filtering, and frequency-selective heat transport applications at microscale.