This article shows how four-dimensional (4D) printing technology can engineer adaptive metastructures that exploit resonating self-bending elements to filter vibrational and acoustic noises and change filtering ranges. Fused deposition modeling (FDM) is implemented to fabricate temperature-responsive shape-memory polymer (SMP) elements with self-bending features. Experiments are conducted to reveal how the speed of the 4D printer head can affect functionally graded prestrain regime, shape recovery and self-bending characteristics of the active elements. A 3D constitutive model, along with an in-house finite element (FE) method, is developed to replicate the shape recovery and self-bending of SMP beams 4D-printed at different speeds. Furthermore, a simple approach of prestrain modeling is introduced into the commercial FE software package to simulate material tailoring and self-bending mechanism. The accuracy of the straightforward FE approach is validated against experimental observations and computational results from the in-house FE MATLAB-based code. Two periodic architected temperature-sensitive metastructures with adaptive dynamical characteristics are proposed to use bandgap engineering to forbid specific frequencies from propagating through the material. The developed computational tool is finally implemented to numerically examine how bandgap size and frequency range can be controlled and broadened. It is found out that the size and frequency range of the bandgaps are linked to changes in the geometry of self-bending elements printed at different speeds. This research is likely to advance the state-of-the-art 4D printing and unlock potentials in the design of functional metastructures for a broad range of applications in acoustic and structural engineering, including sound wave filters and waveguides.