We study the dynamics of a lumped mass linear oscillator that is damped through the use of a material with memory in which the internal dissipative forces depend not only on current but also on previous deformations. This effective memory is governed by two parameters: the relaxation modulus G0, and relaxation time γ, which also govern the vibration-damping properties of the material. Conditions for optimal damping in the unforced case corresponding to critical damping of a linear oscillator with viscous damping are derived, and the response of the oscillator in the case of sinusoidal excitation is studied. When the relaxation time is small the history type damping is modeled approximately by the action of a classical viscous damper with small viscosity. However, when the relaxation time is sufficiently large, this damping mechanism adds to the system a new higher resonance frequency that depends on G0 and γ. Since the oscillator is active over a wide range of frequencies, it has potential applications to the development of adaptive damping devices.