We describe multilayer acoustic nanowave devices based on aperiodic stacks of GaAs and AlAs layers and achievable with standard molecular beam epitaxy (MBE) technology. These nanostructures were designed to display optimized acoustic reflectivity curves in the terahertz range. We discuss the use of different techniques for the design, optimization, and characterization of such acoustic phonon devices. Three optimized acoustic phonon devices were grown by MBE and characterized structurally by x-ray diffraction and photoluminescence: a broadband mirror, a color filter, and an edge filter. The acoustic phonon spectra were studied by Raman scattering in forward and backscattering geometries. We contrast the experimental results with simulations of the Raman spectra using a photoelastic model. We show that Raman spectroscopy provides a powerful tool to acoustically characterize complex aperiodic devices.