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The numerical solution of elliptic or hyperbolic boundary value problems via the Boundary Element Method has a long tradition and is well developed nowadays. The two most popular discretization schemes of the underlying boundary integral equations are the Collocation method and the Galerkin method. While the first one has been adopted to both types of boundary value problems the latter one has been mainly applied to elliptic boundary value problems. To close this gap, the present work is concerned with the derivation of a Symmetric Galerkin Boundary Element Method (SGBEM) for 3-dimensional mixed initial boundary value problems. Thereby, the deduction of the method is presented in an unified manner such that, finally, the scalar wave equation, the system of elastodynamics as well as viscoelastodynamic problems are covered. Contrary to unsymmetric Boundary Element formulations, the SGBEM demands the use of the second boundary integral equation featuring hyper-singularities. With the help of the Stokes theorem those hypersingularities as well as the strong singular integral kernels are transformed into weakly singular integral kernels. Afterwards, the Boundary Element Method is formulated by using standard techniques for the spatial discretization and by applying the Convolution Quadrature Method to the time-convolution integrals. The final numerical tests verify this method and approve its robustness and its reliability. These two properties are an essential prerequisite for a successful use of the proposed Boundary Element Method within a wide range of industrial applications.