The repeating digits in the fractional part are essentially the output of a linear congruence random number generator. Finding the period is almost the same as determining the order of an element of the state space, but by chance you might get a shorter repetition that fits into the longer one.

Finding the order of an element in a group is a hard problem on par with factoring http://mathworld.wolfram.com/GroupOrder.html

I'm not sure about cases where the state space doesn't form a multiplicative group, but I doubt that it gets much easier.