We provide a successful approach towards the solution of the longstanding problem of the large overestimation of the static polarizability of conjugated oligomers obtained using the local density approximation within density-functional theory. The local approximation is unable to describe the highly nonlocal exchange and correlation effects found in these quasi-one-dimensional systems. Time-dependent current-density-functional theory enables us to describe ultranonlocal exchange-correlation effects within a local current description. Recently a brief account was given of the application of the Vignale-Kohn current-functional [G. Vignale and W. Kohn, Phys. Rev. Lett. 77, 2037 (1996)] to the axial polarizability of oligomer chains [M. van Faassen, P. L. de Boeij, R. van Leeuwen, J. A. Berger, and J. G. Snijders, Phys. Rev. Lett. 88, 186401 (2002)]. With the exception of the model hydrogen chain, our results were in excellent agreement with best available wavefunction methods. In the present work we further outline the underlying theory and describe how the Vignale-Kohn functional was implemented. We elaborate on earlier results and present new results for the oligomers of polyethylene, polysilane, polysilene, polymethineimine, and polybutatriene. The adiabatic local density approximation gave good results for polyethylene, which were slightly modified by the Vignale-Kohn functional. In all other cases the Vignale-Kohn functional gave large improvements upon the adiabatic local density approximation. The Vignale-Kohn results were in agreement with best available data from wave function methods. We further analyze the hydrogen chain model for different bond length alternations. In all these cases the Vignale-Kohn correction upon the adiabatic local density approximation was too small. Arguments are given that further improvements of the functional are needed. (C) 2003 American Institute of Physics.