According to unified schemes of Active Galactic Nuclei (AGN), the central engine is surrounded by dusty, optically thick clouds in a toroidal structure. We have recently developed a formalism that for the first time takes proper account of the clumpy nature of the AGN torus. We now provide a detailed report of our findings in a two-paper series. Here we present our general formalism for radiative transfer in clumpy media and construct its building blocks for the AGN problem -- the source functions of individual dusty clouds heated by the AGN radiation field. We show that a fundamental difference from smooth density distributions is that in a clumpy medium, a large range of dust temperatures coexist at the same distance from the radiation central source. This distinct property explains the low dust temperatures found close to the nucleus of NGC1068 in 10 mic interferometric observations. We find that irrespective of the overall geometry, a clumpy dust distribution shows only moderate variation in its spectral energy distribution, and the 10mic absorption feature is never deep. Furthermore, the X-ray attenuating column density is widely scattered around the column density that characterizes the IR emission. All of these properties are characteristic of AGN observations. The assembly of clouds into AGN tori and comparison with observations is presented in the companion paper.
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