This article, the first of a series of two, describes the formulation of Rosenfeld's fundamental measure theory for a mixture of parallel hard cubes, a model recently introduced to study the demixing transition for additive hard core potentials. Special emphasis is put on the good performance of the functional when reducing the dimensionality of the system, a necessary feature to give reasonable results in highly inhomogeneous situations. This property allows for an extremely simple formulation of the theory in arbitrary dimensions. In a subsequent article we will describe the properties of the mixture as they are predicted by the theory, in particular the demixing in presence of the freezing transition. (C) 1997 American Institute of Physics.