Uniform sampling with constraints

Question

Suppose one wants to uniformly sample a string $w$ of a given length over a finite alphabet, such $w$ satisfies a set of structural constraints (such as - "the third character has to be equal to the first character and the last character has to be equal to the second one").
The obvious method is to uniformly sample $w$, check if the constraints hold and return it if so. However, I am dealing with the case where there are many constraints, and the probability of random string to satisfy them is extremely low. Is there a way to sample uniformly from the set of all strings satisfying the set of constraints?