In this paper we propose a new hyper-heuristic that is composed of a simple selection mechanism based on a learning automaton and a new acceptance mechanism, ie the Iteration Limited Threshold Accepting criterion. This hyper-heuristic is applied to the challenging Traveling Tournament Problem. We show that the new hyper-heuristic method consistently outperforms the Simple Random hyper-heuristic even with a small number of low-level heuristics. Moreover, our method, although very general, generates high-quality solutions for the known Traveling Tournament Problem benchmarks and offers new solutions for the recently added Super instances.