A new highly anisotropic coordination heterobimetallic polymer [Mn(III)(Schiff-base)]3[Re(IV)(CN)7] was synthesized and characterized structurally and magnetically. The single crystal X-ray analysis has revealed that this is the first framework among the complexes composed of homoleptic cyanometallate and Mn(III) complex of the tetradentate Schiff base ligand. A formation of 3D assembly is possible due to both the pentagonal bipyrimidal geometry of the cyanometallate unit and suitable size of constituents: [Re(CN)7](3-) and [Mn(III)(acacen)](+), where acacen = N,N'-ethylenebis(acetylacetoneiminato). The powder and crystal magnetic studies show that the compound undergoes an antiferromagnetic ordering of a complicated character below Neel temperature of 13 K, and exhibits a metamagnetic behavior and strong magnetic anisotropy similar to those observed in related 3D Mn(II)-[Mo(CN)7](4-) systems. Unusual magnetic properties of [Mn(III)(acacen)]3[Re(IV)(CN)7] (1) originate from an interplay of Re-Mn anisotropic spin coupling and ZFS effect of Mn(III) ions with a noncollinear orientation of the local magnetic axes in the cyano-bridged 3D network. A theoretical model of anisotropic spin coupling between orbitally degenerate [Re(IV)(CN)7](3-) complexes and Mn(III) ions is developed, and specific microscopic mechanisms of highly anisotropic spin coupling in Re(IV)-CN-Mn(III) linkages in complex 1 are analyzed in detail.