Two-body charmless hadronic B decays involving a light tensor meson in their final states are studied in the perturbative QCD approach based on k(T) factorization. From our calculations, we find that the decay branching ratios for the color-allowed, tree-dominated decays B -> a(2)(0)pi(+) and B -> a(2)(-)pi(+) are of orders 10(-6) and 10(-5), respectively, while other color-suppressed, tree-dominated decays have very small branching ratios. In general, the branching ratios of most decays are in the range of 10(-5) to 10(-8), which are bigger by 1 or 2 orders of magnitude than those predictions obtained in the Isgur-Scora-Grinstein-Wise II model and in the covariant light front approach, but consistent with recent experimental measurements and QCD factorization calculations. Since decays with a tensor meson emitted from vacuum are prohibited in naive factorization, the contributions of nonfactorizable and annihilation diagrams are very important to these decays, which are calculable in our perturbative QCD approach. We also give predictions for the direct CP asymmetries, some of which are large enough for the future experiments to measure. Because we consider the mixing between f(2) and f(2)', the decay rates are enhanced significantly for some decays involving f(2)' mesons, even with a small mixing angle.