This paper presents the construction method of a family of reconfigurable deployable polyhedral mechanisms (RDPMs) based on straight elements. First, reconfigurable straight element (RSE) is designed from two aspects: two prismatic-revolute-revolute-prismatic mechanisms aspect and reconfigurable angulated element aspect, and the kinematics and multifurcation of RSE are investigated. Then reconfigurable multiple straight elements (RMSEs) with n pairs of straight elements are proposed, RMSEs can reach two different transition configurations, the constraint conditions of RMSEs at transition configuration I and transition configuration II are analyzed with all link lengths identified. Finally, two typical polyhedra are used as a basis to construct RDPMs to verify the feasibility of the proposed construction method. A combination of half platforms and whole platforms for the first time is used in the construction of RDPMs, and the obtained mechanisms can switch between two kinds of conventional deployment configurations (the Hoberman sphere motion configuration and radially reciprocating motion configuration) and their compound configurations.