We propose an approach to analytically solve the quantum dynamics of bosonic systems. The method is based on reconstructing the quantum state of the system from the moments of its annihilation operators, dynamics of which is solved in the Heisenberg picture. The proposed dynamical reconstruction method is general in the sense that it does not require assumptions on the initial conditions of the system such as separability, or the structure of the system such as linearity. It is an alternative to the standard master-equation approaches, which are analytically demanding especially for large multipartite quantum systems. To demonstrate the proposed technique, we apply it to a system consisting of two coupled damped quantum harmonic oscillators.