We develop the diagrammatic formulation of the many-body theory for the coupled collective modes in interacting electron systems of different dimensions. The formalism is then applied in detail to a two-dimensional (2D) system coupled to a three-dimensional (3D) electron gas. We find two dimensionally mixed plasmon modes which in the long-wavelength limit are respectively 3D-like and acoustic in nature, but are strongly coupled at larger wave vectors. The same formalism can be applied to any dimensional combinations, and we also present the results for 1D-2D and 1D-3D coupled systems.