Deformable linear objects (DLOs) have a wide variety of applications in a range of fields. Their key characteristic is that they extend much further in one of their dimensions than in the other two. Accurate motion planning is particularly important in the case of DLOs used in robotics applications. In this paper, a new strategy for planning the motions of DLOs under multiple constraints is proposed. The DLO was modeled as Cosserat elastic rods so that the deformation is simulated accurately and efficiently. The control of the motion of the DLO was enhanced by supplementing one gripper installed at each end with additional supports. This allows DLOs to undergo complex deformations, and thus avoid collisions during motion. The appropriate number of supports and their positions were determined, and then a rapidly exploring random tree algorithm was used to search for the best path to guide the DLO toward its target destination. The motion of the simulated DLO is described as it is controlled using two grippers and specific numbers of supports. To prove that the proposed DLO motion planning strategy can successfully guide relatively long DLOs through complex environments without colliding with obstacles, a case study of the strategy was conducted when guiding a DLO through a puzzle.