The 3D printing technology of fused filament fabrication of Lightweight PLA (LWPLA) thermoplastics shows great potential to produce the next generation of reduced weight structural parts. This material is foamed while printing and it allows obtaining a wide range of densities, and consequently, stiffness. Thus, one of the most promising possibilities offered by this material is that it is suitable for manufacturing functionally graded materials. In addition, this material can be used as the core of a sandwich structure, enhancing its lightweight characteristic. Therefore, these sandwiches can be a promising type of panel, as their behaviour, such as the vibracoustic, can be customised. The effectiveness of these laminates in improving vibration performance was not studied, so the applicability of 3D printing technology to manufacture these non-conventional structures for vibration attenuation is a topic to be studied. In the present study, variable-stiffness and variable density 3D printed LWPLA thermoplastic core sandwiches were designed, manufactured, and tested, to improve their dynamic behaviour. The LWPLA was characterized to obtain its Poisson’s ratio, elasticity modulus and shear modulus as a function of the part density, which is controlled by the temperature and the feed rate of the printing process. A finite element model of a sandwich panel with spatially variable density and mechanical properties was defined for the optimisation of the design. A genetic optimisation algorithm was used to obtain the optimum values of the material properties of the LWPLA core of the sandwich that maximises the first two natural frequencies for a plate with one embedded edge and three free edges. The most promising sandwiches were manufactured and tested. The functionally graded core sandwiches showed a great improvement of the natural frequencies. This study is a significant contribution to the design and manufacturing of 3D printed LWPLA core sandwich panels, and an additional step towards producing high-performance functionally graded components for improved vibroacoustic behaviour.