The new calcium iron arsenide compounds Can(n+1)/2(Fe1-xMx)((2+3n))M' As-n(n-1)/2((n+1)(n+2)/2) (n = 1-3; M = Nb, Pd, Pt; M' = square Pd, Pt) were synthesized and their crystal structures determined by single-crystal X-ray diffraction. The series demonstrates the structural flexibility of iron arsenide materials, which otherwise prefer layered structures, as is known from the family of iron-based superconductors. In the new compounds, iron arsenide tetrahedral layers are bridged by iron-centered pyramids, giving rise to so far unknown frameworks of interconnected Fe As layers. Channels within the structures are occupied with calcium and palladium or platinum, respectively. Common basic building blocks are identified that lead to a better understanding of the building principles of these structures and their relation to CaFe4As3.