Bubbles are highly efficient sound scatterers at their resonance frequencies. When present in sufficient quantities, they induce significant frequency-dependent variations in sound speed, nonlinear parameter, and attenuation. This property allows for the manipulation of acoustic signals using bubble populations, which can be introduced into a medium through gas injection or contrast agent microbubbles. The ability to control sound using oscillating gas bubbles led to the concept of bubbly-liquid-based acoustic metamaterials (BLAMMs). One of the main challenges is achieving uniform bubble size and ensuring their stability within liquids. To address these issues, replacing the liquid medium with viscoelastic media presents a more practical solution. In this study, we investigate the nonlinear propagation of ultrasonic waves in air–viscoelastic mixtures with nonhomogeneous bubble distributions. The presence of bubble layers or clouds introduces acoustic screening effects that modify ultrasonic wave behavior. One-dimensional plane waves are considered, where bubble vibrations interact with the acoustic field through a coupled system: nonlinear Rayleigh–Plesset equation for bubble dynamics and the linear, non-dissipative wave equation for sound propagation. We focus on the filtering effects of bubble layers at specific frequency ranges, examining phenomena such as acoustic shielding and resonance. The influence of bubble spatial distribution on nonlinear wave propagation is analyzed, considering parameters like bubble density, size, layer position, and interlayer distances. Additional factors—such as source-layer distance, number of layers, and acoustic pressure characteristics—further influence the filtering effect. Finally, the role of viscoelasticity is also investigated with the aim of using bubbly elastic structures in experimental metamaterial design. Numerical simulations are performed in both linear and nonlinear excitation regimes, revealing the impact of bubbles on acoustic wave propagation. The model proves to be an efficient tool for characterizing and designing nonlinear acoustic systems, demonstrating the potential of bubbles in acoustic metamaterial development.