Vibrations are undesirable and cause many problems in engineering. Among the many techniques to reduce vibrations, phononic crystals and elastic metamaterials, that have periodic variations in geometry or material properties, have the potential to attenuate vibrations over a large range of frequencies. These classes of materials attenuate vibrations via the band gap mechanism that stops elastic waves from propagating. Additionally, periodic structures can possess high effective stiffness or strength to density ratios. This makes them ideal for lightweight structural applications. Furthermore, their periodic nature allows band gaps to be generated in them. These features can be combined to produce lightweight structural components with vibration isolation properties. Currently, very few studies of this class of materials for practical applications were made. Therefore, the goal of the work done in this thesis is to develop techniques and perform analyses to promote their implementation. A literature review has been performed on the techniques to determine the band structures and effective properties of lattice materials. A novel method coined the wave superposition method (WSM) to measure the band structure was developed. This method allows band structures to be determined experimentally using simple equipment with a small number of measurements. The method was then validated experimentally. A parametric study on the mechanical and dispersion properties of cubic lattice structures were conducted to assess the viability of designing a multifunctional lattice structure with excellent properties to be used as multifunctional lightweight and vibration attenuating components. Important trends relating to the geometric parameters to the performance of the lattice structures were found. Lastly, an experimental study was performed on a selected design to demonstrate the vibration attenuation characteristic of this material. The techniques and findings in this thesis, have laid the foundations for future development of periodic structures for both structural and vibration isolation applications.