Conjugated polymers are interesting for their large nonlinear optical properties. Time-dependent density-functional theory is a very versatile method to calculate such properties of large systems. However, even the linear axial polarizability of conjugated oligomers is greatly overestimated when using the standard adiabatic local density approximation (ALDA). This approximation, but also more advanced approximations, is unable to describe the highly nonlocal exchange-correlation found in these systems. We use time-dependent current-density-functional theory, in which we describe this ultranonlocal exchange-correlation within a local current description. The static axial polarizability of polyacetylene (see figure), and similar polymeric systems, obtained using the Vignale-Kohn (VK) [G. Vignale and W. Kohn, Phys.Rev.Lett. 77, 2037 (1996)] current-functional is in excellent agreement with MP2 results. For these systems and a set of benchmark molecules we obtained the excitation energies and oscillator strengths, and from these we explain the large reduction of the static polarizability by the VK-functional.