Density-functional theory (DFT) calculations of indirect nuclear magnetic resonance spin-spin coupling tensors J, with the anisotropic but symmetric parts being the particular concern, are carried out for a series of molecules with the linear response (LR) method. For the first time, the anisotropic components of J are reported for a hybrid functional. Spin-spin tensors calculated using the local density approximation (LDA), the gradient-corrected Becke-Lee-Yang-Parr (BLYP) functional, and the hybrid three-parameter BLYP (B3LYP) functional are compared with previous ab initio multiconfiguration self-consistent-field (MCSCF) LR results and experimental data. In general, the B3LYP functional provides reasonable accuracy not only for the isotropic coupling constants but also for the anisotropic components of J, with the results improving in the sequence LDA-->BLYP-->B3LYP. Error cancellation often improves the total DFT spin-spin coupling when the magnitude of the paramagnetic spin-orbit contribution is overestimated, or when the spin-dipole (SD) and Fermi-contact (FC) contributions are far from the MCSCF values. For the F-19 nucleus, known to be difficult for DFT, the anisotropic properties of heteronuclear, in particular (FC)-F-19-C-13 couplings are often more accurate than the poorly described isotropic coupling constants. This happens since the FC contribution is small at fluorine compared with carbon, leading to a small error in the total SD/FC term. With the recent implementation of the hybrid B3LYP functional, calculations of predictive quality for the J tensors are no longer restricted to small model molecules, opening up the possibility of studying the anisotropic components of J in large organic and biomolecules of experimental interest.