The effect of rotation and vibration on the nuclear magnetic resonance (NMR) shielding constants was computed for HF, F-2, N-2, CO, and HBr. The shielding constants for H, C, N, O, and F nuclei were calculated using sum-over-states density functional perturbation theory (SOS-DFPT). Diatomic ro-vibrational states were calculated from a discrete variable representation using Morse potentials and potential curves calculated with density functional theory. Our ro-vibrational corrections to shielding constants for HF, CO, F-2, and N-2 molecules are in good agreement with experimental data and CCSD(T) calculations. These results together with satisfactory first and second derivatives of the shielding constants with respect to interatomic distances confirm that the shielding surfaces produced by the SOS-DFPT method are of good accuracy, providing reassurance of the use of these methods for more complex systems. The unusual temperature dependence of the hydrogen chemical shift in HBr and a first attempt to include both relativistic spin-orbit and ro-vibration effects are discussed.