In order to make efficient use of the available luminosity and to protect detector components, detectors at the future linear collider must be kept tolerably free of accelerator-related backgrounds. This requires the use of mechanical collimators that intercept particles with large transverse amplitudes while transmitting the core of the beam to the collision point. Since collimators must be placed very close to the beam line to achieve the desired level of background suppression, the wakefields induced in collimators by the passing beam act back on the beam particles and cause them to deviate from their original trajectories, resulting in jitter amplification and emittance dilution. This, in turn, reduces the luminosity, increases detector backgrounds, and results in machine protection issues. The ability to reliably predict and reduce collimator wakefields is therefore critical for optimizing the linear collider design. We will describe the facility that has been built at SLAC Linac for precise measurements of short-range transverse wakefields in collimators of various geometries and materials, present preliminary results of measurements with the first three collimator sets, and compare the results with existing analytical models and numerical simulations. Current status of the experiment and future plans will be discussed.