This article presents a study on the use of the dynamical response of multi-laminated periodic bars to create resonance band gaps within useful frequency ranges. The objective is to control, in a passive form, the longitudinal vibration transmissibility in specific and wide enough frequency ranges of interest. This is achieved by the separation of two adjacent eigenfrequencies. A relation between the modal analysis, the harmonic analysis and the Bloch wave theory is proposed, for which no reference was found in the searched literature. As shown, the selection of appropriate material pairs is essential to obtain useful frequency ranges. The use of pairs of steel and cork agglomerate is proposed, since it allows the design of attenuators at lower frequencies through a prediction based on finite element analysis (FEA). This approach requires the storage modulus of cork for which analytical and numerical FEA models were verified and validated. A methodology to obtain experimentally the storage modulus of cork is presented. Regarding the structural improvement problem, we discuss a methodology to design periodic bars for a specific location of the first attenuation's frequency range and illustrate the main results through several examples.