The band electronic structures of one-dimensional polymers composed of thiophene, pyrrole, and tricyclic nonclassical thiophenes ([1,2,5]thiadiazolo[3,4-b]thieno[3,4-e]pyrazine and dithieno[3,4-b:3',4'-e]pyrazine) are calculated and analyzed at the extended Huckel level of theory, with the development of highly conducting polymers in mind. The tricyclic nonclassical thiophenes that can impose quinoid-type characters to the resulting polymers are effective building blocks for the preparation of small band-gap polymers. Calculated band gaps are discussed in view of the frontier crystal orbitals and the bond length alternation of the polymers. The homopolymer of [ 1,2,5]thiadiazolo[3,4-b]thieno[3,4-e]pyrazine that is predicted to have a small band gap of 0.1 eV is a good candidate for an intrinsic conducting polymer without dopants.