For any positive odd integer d not divisible by 3, define the arithmetical function $T_d(m)$ equal to $mover2$ if m is even, and $3m+d}over2$, otherwise. The $3x+d$ hypothesis, generalizing the well-known $3x+1$ conjecture, asserts that the dynamical system generated by the function $T_d(m)$ has a finite number of cycles and no divergent trajectories. We study here the cyclic structure of this system, and prove in particular an effective polynomial upper bound to the number of cycles with a given number of odd members.