In this work we study the homotopy type of multipath complexes of bidirectional path graphs and polygons, motivated by works of Vrećica and Živaljević on cycle-free chessboard complexes (that is, multipath complexes of complete digraphs). In particular, we show that bidirectional path graphs are homotopic to spheres and that, in analogy with cycle-free chessboard complexes, multipath complexes of bidirectional polygonal digraphs are highly connected. Using a Mayer-Vietoris spectral sequence, we provide a computation of the associated homology groups. We study T-operations on graphs, and show that this corresponds to taking suspensions of multipath complexes. We further discuss (non) shellability properties of such complexes, and present new open questions.