The unsteady (oscillatory) Stokes resistance is numerically computed for finite‐length, circular cylinders using a first‐kind, boundary integral formulation for low and moderate dimensionless frequencies. Translational oscillations parallel and perpendicular to the cylinder axis (axial and transverse oscillations) are considered. For arbitrarily oriented cylinders with length‐to‐width ratios in the range: 1/10≤a/b≤10, a simple, previously proposed, expression accurately (±5%) describes the oscillatory Stokes resistance at all frequencies in terms of steady Stokes and inviscid flow resistance parameters. Numerical calculations for spheroids agree with previous results; the results for cylinders and transverse oscillations of oblate spheroids are new.