There is a commonly held opinion that the algorithms for learning unrestricted types of Bayesian networks, especially those based on the score+search paradigm, are not suitable for building competitive Bayesian network-based classifiers. Several specialized algorithms that carry out the search into different types of directed acyclic graph (DAG) topologies have since been developed, most of these being extensions (using augmenting arcs) or modifications of the Naive Bayes basic topology. In this paper, we present a new algorithm to induce classifiers based on Bayesian networks which obtains excellent results even when standard scoring functions are used. The method performs a simple local search in a space unlike unrestricted or augmented DAGs. Our search space consists of a type of partially directed acyclic graph (PDAG) which combines two concepts of DAG equivalence: classification equivalence and independence equivalence. The results of exhaustive experimentation indicate that the proposed method can compete with state-of-the-art algorithms for classification.