You might be interested in this paper on discovering cyclic causal models: http://arxiv.org/abs/1206.3273
While cycles can be introduced into directed graphical models, it makes it significantly more complicated to compute the probability of some configuration. For the graphical model $X\rightarrow Y$, if we know the marginal probability of $X$ and the conditional probability of $Y$ given $X$, then the joint probability of $X=x$ and $Y=y$ is just $Pr(X=x,Y=y) = Pr(X=x)Pr(Y=y|X=x)$, and in general any probability can be computed by a combination of multiplication and marginalization. If by contrast we have a cyclic graphical model $X \leftrightarrow Y$, then this is best viewed as a Markov chain $\ldots X\rightarrow Y \rightarrow X \rightarrow Y \ldots$ where, even if we know the transition probabilities $p(x|y)$ and $p(y|x)$, we still have to solve for the stationary state of a Markov chain to compute various probabilities. For cyclic models with larger graphs the situation becomes even more complicated.
I realize this doesn't solve your problem exactly though. Spelling out the problem you're trying to solve in more detail might help. For instance, if the influences between A, B and C are symmetric (A influences B the same way B influences A) then an undirected graphical model would make more sense. If there is a temporal aspect to the influence of one variable on another, a Markov chain would make more sense (and in some sense a cyclic causal model can be seen as a kind of Markov chain, where we are trying to reason about properties of the stationary distribution).