By using the theory of D-modules we express the characteristic cycle of a local cohomology module supported on a monomial ideal in terms of conormal bundles relative to a subvariety. As a consequence we can decide when a given local cohomology module vanishes and compute the cohomological dimension in terms of the minimal primary decomposition. We can also give a Cohen-Macualayness criterion for the quotient of a polynomial ring by a monomial ideal and compute its Lyubeznik numbers.