High-resolution nuclear resonance fluorescence experiments (NRF) were performed on 110,111,112,114,116Cd at the bremsstrahlung facility of the 4.3-MV Dynamitron accelerator in Stuttgart to study the low-lying dipole strength distributions in these vibrational nuclei. Numerous excited states, most of them previously unknown, were observed in the excitation energy range up to 4 MeV. Detailed spectroscopic information has been obtained on excitation energies, spins, decay widths, decay branchings, and transition probabilities. For states in the even-even isotopes 110,112,114,116Cd, parities could be assigned from linear polarization measurements. Together with our previous results for 108,112,113,114Cd from NRF studies without polarization measurements, systematics was established for the dipole strength distributions of the stable nuclei within the Cd isotopic chain. The results are discussed with respect to the systematics of E1 two-phonon excitations and mixed-symmetry states in even-even nuclei near the Z=50 shell closure and the fragmentation of these excitation modes in the odd-mass Cd isotopes.