Exchange-only ab initio (parameter-free) time-dependent density functional calculations for the vertical excitation energies of atoms and polyatomic molecules are performed by employing optimized effective potentials (OEP's) and their corresponding adiabatic exchange kernels for the first time. Accurate OEP's are obtained by a novel linear-combination-of-atomic-orbital (LCAO) algorithm [R. Colle and R. K. Nesbet, J. Phys. B 34, 2475 (2001)] in which a potential is represented as a sum of a seed potential having the correct -1/r asymptotic behavior and a small and rapidly decaying correction, the latter being approximated accurately by a linear combination of Gaussian functions. The time-dependent OEP (TDOEP) methods with and without the Tamm-Dancoff approximation are implemented by using a trial-vector algorithm, which allows us to avoid the storage or manipulation of transformed two-electron integrals or the diagonalization of large matrices. No approximation is made to TDOEP, besides the adiabatic approximation to the exchange kernel, the LCAO expansion of the orbitals and potentials, and occasionally the Tamm-Dancoff approximation. The vertical excitation energies of the beryllium atom and the nitrogen and water molecules calculated by TDOEP are compared with those obtained from time-dependent density functional theory (TDDFT) employing conventional local or gradient-corrected functionals, configuration interaction singles (CIS), time-dependent Hartree-Fock (TDHF) theory, similarity-transformed equation-of-motion coupled-cluster with single and double substitutions, and experiments. TDOEP, which neglects electron correlation while treating the exchange contribution rigorously within the Kohn-Sham DFT framework, performs equally well as, or even appreciably better than, CIS or TDHF. The slightly better performance of TDOEP might be attributed to the local nature of the exchange potentials that allows the bare orbital energy differences to approximate excitation energies well. Nevertheless, TDDFT employing local or gradient-corrected functionals outperforms TDOEP for low-lying valence excited states, implying that the former somehow accounts for electron correlation effectively, whereas for high-lying and Rydberg excited states, the latter performs better than the former. By combining the desirable features of OEP and local or gradient-corrected exchange-correlation potentials, we arrive at a simple asymptotic correction scheme to the latter. TDDFT with the asymptotic correction yields uniformly accurate excitation energies for both valence and Rydberg excited states.