Three-dimensional (3D) halide perovskites have attracted enormous research interest, but the choice of the A-site cations is limited by the Goldschmidt tolerance factor. In order to accommodate cations that lie outside the acceptable range of the tolerance factor, low-dimensional structures usually form. To maintain the favorable 3D connection, the links among the metal-halide octahedra need to be rearranged to fit the large cations. This can result in a departure from the proper corner-sharing perovskite architectures and lead to distinctly different perovskitoid motifs with edge- and face-sharing. In this work, we report four new 3D bromide perovskitoids incorporating linear organic diammonium cations, A’Pb(2)Br(6) (A’ is a +2 cation). We propose a rule that can guide the further expansion of this class of compounds, analogous to the notion of Goldschmidt tolerance factor widely adopted for 3D AMX(3) perovskites. The fundamental building blocks in A’Pb(2)Br(6) consist of two edge-shared octahedra, which are then connected by corner-sharing to form a 3D network. Different compounds adopt different structural motifs, which can be transformed from one to another by symmetry operations. Electronic structure calculations suggest that they are direct bandgap semiconductors, with relatively large band dispersions created by octahedra connected by corner-sharing. They exhibit similar electronic band structures and dynamic lattice characteristics to the regular 3D AMX(3) perovskites. Structures with smaller Pb-Br-Pb angles and larger octahedra distortion exhibit broad photoluminescence at room temperature. The emerging structure-property relationships in these 3D perovskitoids set the foundation for designing and investigating these compounds for a variety of optoelectronic applications.