Undeterred by the fact that the 1-click root method available for other Motorola devices doesn't work for the retail version of Verizon's Droid RAZR, the Android community has once again liberated the latest addition to the Droid family. Droid Forums is crediting Pieman13 for the find, providing an easy 5-step guide to rooting your RAZR. Before we go any further, I should encourage those unfamiliar with rooting to check out our primer on the subject here. If your phone gets bricked, there aren't any unbricking methods just yet, so proceed with caution. Of course Android Police claims no responsibility for what might happen to your device.
Without further ado, here are the steps to get your brand new Droid RAZR rooted:
Step 1: Download and install Motorola Drivers for your Windows PC (this doesn’t work on other operating systems): 32-bit or 64-bit.
Step 2: Download DooMLoRD v2 ROOT zergRush busybox su.zip and extract contents to your PC (desktop would be a good place).
Step 3: From your Droid RAZR’s homescreen, tap Menu button and enable USB Debugging from Settings > Applications > Development.
Step 4: Connect your Droid RAZR with your PC and select Charge Only option from notifications drawer.
Step 5: Navigate to where the folder you unzipped in Step 2 to and open the folder and click on the runme.bat file and let it do its thing.
And there you have it! If all went well, your RAZR should now be rooted and running Android 2.3.5. Gingerbread.
Source: Droid Forums