The main interest of this study is to analyze the effect of Poisson’s ratio (negative for auxetic materials) on the dynamic behavior of anti-trichiral structures: an experimental and numerical study is proposed. Using three-dimensional printing technology, anti-trichiral structures (cores and sandwiches) with different geometric parameters were produced with a bio-based composite material. The material used was polylactic acid (PLA) with flax fibers. The structure geometry (anti-trichiral) is the main factor that has an influence in determining its auxetic behavior. In this study, we show the effect of the elementary cell size of the anti-trichiral structure on the auxetic behavior and its impact on the static and dynamic properties of the materials. Initially, experimental tensile tests were used to establish the static characteristics of all specimens, including Poisson’s ratio values. From these results concerning Poisson’s ratios, experimental vibration tests were also carried out to determine their dynamic properties and this is to understand the field of application of these structures and to analyze their behavior in practical use. To validate experimental results, finite element studies were carried out. In this study of the dynamic properties, we observe close agreement convergence between the results of the experimental tests and the numerical analysis.