The chemical bonding in an intriguing hypothetical organometallic three-dimensional structure, the realization of a (3,4)-connected net, is analyzed by band-structure calculations. The net, stoichiometry FeC4, has a structure which consists of infinite linear chains of Fe atoms that are cross-linked through kinked polyacetylene-like chains of carbon atoms. Densities of states and calculations (using the extended Huckel approximate molecular orbital method) of a number of related one-dimensional infinite and molecular models are used to describe the bonding characteristics in the carbide. When combined in a three-dimensional framework, carbon and metal chains do not show any tendency to undergo the simple pairing distortions which are characteristic of isolated metal and carbon chains. That these separated chains lose their driving force toward distortion upon fusion is explained by charge transfer from the carbon pi-system to the metal d block and a more rigid sigma-bonded framework. If one moves to MC(4) with M carrying fewer electrons than Fe, the MM bond strengthens, and CC bonding is weakened; for M having more electrons the trend is reversed. A number of intriguing hypothetical related compounds, formed by incorporation of metal chains into the large channels present in the MC(4) phase or by substitution of polyacenes for polyacetylenes, are proposed.