We present an adaptation of the so-called structural method [6] for Hamiltonian systems, and redesign the method for this specific context, which involves two coupled differential systems. Structural schemes decompose the problem into two sets of equations: the physical equations, which describe the local dynamics of the system, and the structural equations, which only involve the discretization of a very compact stencil. They have desirable properties, such as unconditional stability and high-order accuracy [7]. We first give a general description of the scheme for the one-body case (which corresponds to e.g. spring-mass interactions or pendulum motion), before extending the technique to the multi-body case (treating e.g. the n-body system). The scheme is also written in the case of a non-separable system (e.g. a charged particle in an electromagnetic field). We prove that the method is exactly energy-preserving for quadratic Hamiltonians, and we give numerical evidence of the method's efficiency, its capacity to preserve invariant quantities such as the total energy, and draw comparisons with the traditional symplectic methods.