The LZ dark matter detector, like many other rare-event searches, will suffer from backgrounds due to the radioactive decay of radon daughters. In order to achieve its science goals, the concentration of radon within the xenon should not exceed 2 μBq/kg, or 20 mBq total within its 10 tonnes. The LZ collaboration is in the midst of a program to screen all significant components in contact with the xenon. The four institutions involved in this effort have begun sharing two cross-calibration sources to ensure consistent measurement results across multiple distinct devices. We present here five preliminary screening results, some mitigation strategies that will reduce the amount of radon produced by the most problematic components, and a summary of the current estimate of radon emanation throughout the detector. This best estimate totals < 17.3 mBq, sufficiently low to meet the detector's science goals.