Decidable problems for which no concrete decision procedure is known
Here is a trivial example of such a problem: consider any yes-no question with an unknown answer, e.g. $P=NP$. For any $n \in \mathbb{N}$, let $A(n)$ be true iff $P=NP$. Then $A$ is decidable by one of two decision procedures: 


Always return "true".
Always return "false".


But you need to solve $P=NP$ to know which one.