The Type II error corresponds to rejecting the alternative when you should have retained it.
Basically you need to have a distribution for the alternative (which will be a binomial or a normal approximation given enough data) and you need to find the probability of rejecting the alternative given that its true.
In other words, find the complementary probability where theta > mu and this will be your Type II error in the same way that theta != mu (for a two-sided test) will give you your Type I error.
Its a lot easier to understand if you think in terms of each hypothesis test having its own distribution, test statistic and probability value. Trying to understand it in any other way usually leads to confusion.