Ensure you have backed up all of your photos etc first and that you have synced your contacts with google etc.
I have successfully installed Cyanogen mod 10.10.1 on my Samsung Galaxy S3. Everything seems to work.
The phone had already been registered with Ting and used for months with stock firmware but I wanted to get rid of the unwanted "extras" Samsung pre installed.
I followed these instructions:
With the following rom (I chose 'stable'):
Additionally I downloaded google apps from here:
I used Ubuntu.
I downloaded the 32 bit version of Heimdall (despite being on a 64bit machine).
For the adb steps, I simply:
sudo apt-get install android-tools-adb
The instructions for "adb push" are lacking in detail. The process is (sudo is important):
sudo adb start-server
sudo adb push cm-10.1.1-d2spr.zip /sdcard/
sudo adb push gapps-jb-20130301-signed.zip /sdcard/
Hope that helps.