CyanogenMod is Dead as of 31/12/2016 but the code of CyanogenMod has gave birth to Lineage OS and its just a branding change with few tweaks and not to worry as soon there will be lot of Official ROM builds for many devices and it work pretty much the same as CyanogenMod if not better.
The Official Lineage OS ROM has just been started rolling out and Galaxy S3 has got its official build. Below are the instructions on how to flash with download links.
Install Lineage OS on Samsung Galaxy S3 (verizon)
1) You Need to have a Unlocked bootloader and have a custom recovery like TWRP flashed.
2) Take a Complete Backup of your Device. This is the most important step as if you screw up something you'll have a complete backup so you can restore to the previous state. Here is a video on How to Backup your android.
4) Download Gapps for google apps from here.
5) Copy the downloaded files to the internal storage of your Samsung Galaxy S3.
6) Shut down your Android device and then hold Volume Down and Power button simultaneously to boot into recovery.
8) Go back to the home page of the TWRP then tap on -->"Install", then navigate to the folder where you have copied the Lineage OS rom and the Gapps file which you downloaded before.
9) Select the -->"Lineage OS rom zip file" and then click on -->"Add More Zip Files", then select the -->"Open Gapps Zip File", then swipe to confirm the flash and wait until it gets completed. Have patience as it may take 5-20 minutes for installation.
11) Now select your Language and configure your device like you would do when you have brought an new Android.
12) While configuring your google account if you are redirected back to the first stage of configuration then disconnect from the wifi and turn of your Mobile data and skip google account configuration for now and you will be good to go!
TWRP for Samsung Galaxy S3 (i9300) :- https://dl.twrp.me/d2vzw/
Lineage OS for Samsung Galaxy S3 :- https://download.lineageos.org/d2vzw
Gapps for Samsung Galaxy S3 :- http://opengapps.org/
I didn't Install this as I don't have an Samsung Galaxy S3 device and the Images may be different for you but the procedure will be 99% same below is the video where i install lineage OS on OnePlus 2 its just for reference. if you have any problem watch the video.