A new rocking constraint device (RCD) is developed for three-dimensional (3D) base-isolated frame structures by connecting a custom-designed cylinder pair to provide vertical damping with replaceable damping components installed outside the cylinders when the superstructure undergoes translational motion, and rocking constraint capacity when the superstructure is susceptible to rocking. Theoretical formulas for calculating the damping and rocking constraint stiffness of the RCD are proposed. Two series of sinusoidal loading tests are conducted at different loading frequencies and amplitudes to verify the damping and rocking constraint performance of the RCD. The test results show that the cylinder without orifices on its piston can provide the desired damping with a replaceable damping component, and that the RCD can effectively suppress rocking. Although the vertical stiffness of an individual cylinder is affected by the location of the replaceable damping component and loading frequency, the average vertical stiffness of the two cylinders, which determines the rocking constraint stiffness of the RCD, is independent of the two factors. Comparisons of the test and theoretical results indicate that the errors of the proposed formulas for calculating the damping and rocking constraint stiffness of the RCD do not exceed 12.9% and 11.0%, respectively.