We present a modified configuration of a tunable Alvarez lens with a refocusing frequency of 1 kHz or more. In contrast to the classic Alvarez lens, the approach does not utilize a translational motion of two sub-lenses with respect to each other, but uses a 4f-setup to image two diffractive sub-lenses onto each other. Hereby focus tuning is achieved by rotating a galvo-mirror which affects the overlap of the two sub-lenses which together form an effective lens of refractive power which depends on the rotation angle of the galvo-mirror. We have demonstrated tuning of the optical power in a system where the diffractive Alvarez lens is realized by an LCOS-SLM. We consider our Alvarez setup especially suitable for applications where high refocusing rates are important, as for example in 3D life cell monitoring or tracking.