We have calculated the anisotropic magnetic properties of monoatomic chains of iron atoms embedded in fcc copper. The calculations are based on a relativistic extension of the locally self-consistent multiple scattering method that is able to treat the large supercells required to model inhomogeneous systems. We have investigated two chain geometries: Fe chains along the 110 and 100 directions. We found that the magnetocrystalline anisotropy energy favored orientations of the magnetic moments perpendicular to the chains, while the magnetostatic energy was lowest when the moments are aligned parallel to the chain. Interestingly, our parameter free qualitative results implies that these competing effects are finely balanced.