Many real-world constraint satisfaction problems are structured, i.e., constraints are not uniformly distributed among the set of variables. This structure may be used to improve the solution process of these problems. In particular, backtracking with tree decomposition (BTD) exploits the structure to define variable ordering heuristics and to learn structural goods and nogoods which are used to avoid redundant explorations. BTD is based on a chronological backtracking search. Our goal in this paper is to investigate the interest of exploiting structure when using other approaches for exploring the search space: other complete search approaches, such as conflict directed backjumping (CBJ), but also incomplete approaches, such as decision repair (DR). To this aim, we describe a generic framework for solving CSPs, which is an extension of the framework proposed by Pralet and Verfaillie in [PV04]. Using our new generic framework, we reformulate some existing search procedures, including BTD. We also describe new search procedures which combine structural (no)goods with CBJ and DR. This generic framework allows us to experimentally evaluate the interest of exploiting the structure for different kinds of search procedures.