Technically no, you don't have to root your current ROM before you can flash CyanogenMod on your S3. Instead, you will simply gain root when you flash CyanogenMod itself.
This is due to the way in which Samsung's bootloaders work in comparison to the ones on other devices. Samsung devices effectively have unlocked bootloaders right out of the gate, but they don't have a way to use the fastboot utility that other devices use. Instead, you can use the Odin utility (and a third party, cross platform implementation: Heimdall) to reflash the device from its "Download Mode" at any time, as the CM installation instructions describe.
Therefore, since Download Mode is accessible to the end user without any sort of modifications to the device, you don't need to "root" beforehand in order to install a custom recovery (and, by association, custom ROMs). The reason why you need root on many other devices is because it's the only way to write to the recovery partition and get the custom recovery installed. Often this requires unlocking the bootloader in the process as well.