  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 10\mic\
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.
