We study a coarse-grained protein model whose primary characteristics are (i) a tubelike geometry to describe the self-avoidance effects of the polypeptide chain and (ii) an energy function based on a one-dimensional structural representation. The latter specifies the connectivity of a sequence in a given conformation, so that the energy function, rather than favoring the formation of specific native pairwise contacts, promotes the establishment of a specific target connectivity for each amino acid. We show that the resulting dynamics is in good agreement with both experimental observations and the results of all-atoms simulations. In contrast to the latter, our coarse-grained approach provides the possibility to explore longer time scales and thus enables one to access, albeit in less detail, larger regions of the conformational space. We illustrate our approach by its application to the villin headpiece domain, a three-helix protein, by studying its folding behavior and determining heat capacities and free-energy landscapes in various reaction coordinates.