Multiple mobile assemblies have been created based on two-dimensional tessellations of linkages for deployable structures. However, few three-dimensional tessellations of linkages have been created, especially mobile assemblies with bifurcation. Here, we proposed four types of mobile assemblies of kaleidocycles, a special type of threefold-symmetric Bricard linkages, based on cubic cellulation and symmetry. Kinematic analysis of them is carried out based on the matrix method and numerical method. Two assemblies have bifurcations with two motion paths following cuboid symmetry and tetrahedral symmetry, respectively. Meanwhile, the other two have one motion path with single degree of freedom. The designing process facilitates the creation of new mobile assemblies under symmetry and the four assemblies have the potential application for designing metamaterials.