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 Official ROM builds will work pretty much the same as CyanogenMod if not better.
The Official Lineage OS ROM has just been started rolling out and Redmi Note 3 has got its official build. Below are the instructions on how to flash with download links.
Install Lineage OS on Xiaomi Redmi Note 3
1) You Need to have a Unlocked bootloader and have a custom recovery like TWRP flashed. If you haven’t already done then there is an article on How to flash TWRP and root Redmi Note 3 read it here
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.
3) Download Lineage OS for Redmi Note 3 From Here.
4) Download Gapps for google apps from here.
5) Copy the downloaded files to the internal storage of your Redmi Note 3.
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!
Links:-
Unlock bootloader and flash TWRP :- http://bit.ly/2j37Mb4
TWRP for Redmi Note 3 :- https://dl.twrp.me/kenzo/
Lineage OS for Redmi Note 3 :- https://download.lineageos.org/kenzo
Gapps for Redmi Note 3 :- http://opengapps.org/
I didn’t Install this as I don’t have an Redmi Note 3 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.
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Followed the steps to boot 'lineage-14.1-20170913-nightly-kenzo-signed' on my Xiaomi redmi note 3.
installed lineageOS and GApps successfully (system skipping 'md5 not found' warning itself).
After pressing 'Reboot System', it's stuck at boot screen (black screen with moving O). waited for approx 50 minutes.. still stuck there. Please help.
This could happen for a number of reasons, so as I understand your phone is stuck in the lineage OS Boot animation.
*First Just try holding down the power button for about 10+ seconds until the screen turns off now turn it on again, now it my reboot by itself 2-5 times and again the boot animation will appear juat wait for 15 minutes or so it should bring you the android setup procedure.
*If it still doesn't work try going back to recovery mode after you've turned off the device. Now connect this to your computer then download and copy over the latest TWRP recovery version available to your phone and then flash this image (having the latest recovery version is crucial). Now wipe all the partitions other than internal storage as shown in this guide and flash a Fresh version of latest Lineage OS. But make sure you're using the right version of Gapps if you use a different version, platfrom or anything else you'll again face bootloops and other things. I belive Redmi Note 3 Uses a 64bit processor so you'll have to download 'ARM64' 'Android 7.1' gapps packages from opengapps.org i'd recommend you to go for Nano or the Micro packages there.
* try flashing once without Gapps if nothing works.
* As a Final resort you can try resetting back to original state of your device using a guide like this http://en.miui.com/thread-235865-1-1.html and then starting over with the process!
Thanks Anvith for your reply. I am doing this for the first time so may be I have missed something or did something wrong. Here is the sequence of activities I have followed:
1. Flashed my device by MI Flash using
'kenzo_global_images_V7.1.8.0.LHOMICL_20160129.0000.14_5.1_global'.
Selected 'Flash All'. This completed successfully.
2. Unlocked bootloader successfully by connecting in fastboot mode.
3. Somehow the phone was Rebooted normally and I noticed that it's not detecting the SIM (I thought this
will be fine once I install LineageOS14.1 so didn't bother much). I also noticed that USB
debugging was disabled automatically. So I enabled USB debugging and rebooted to fastboot again.
4. Flashed TWRP (twrp-3.1.1-0-kenzo.img)
5. Rebooted the Phone in recovery mode by pressing and holding 'Power+Volume UP' buttons.
6. Wiped Data,System,Cache, Dalvik/ART Cache and Internal Memory (Tried all the steps again without
wiping internal memory but didn't work)
7. Installed 'lineage-14.1-20170913-nightly-kenzo-signed.zip' and 'open_gapps-arm64-7.1-stock-
20170924.zip'. Both were successful.
8. Tried to reboot the device normally. Black screen with MI written on center appeared then blue Lineage
OS logo appeared with moving O. I waited for more than an hour but no Luck. It was still with moving O.
Phone was heating up so I entered in recovery again and did 'Power Off'.
Please note that I tried to install 'SR4-SuperSU-v2.82-SR4-20170918204844.zip' and
'miui_HMNote3ProGlobal_7.9.22_Firmware.zip' also as suggested in some of the posts about firmware but no Luck. I tried flashing without GApps also, but same issue. Now I doubt the Global Stable Kenzo ROM that I have used but not sure. Suggest if you got any idea about it.
seems like you've done it right but even i cant figure out why this is happening.
But I did some searching and also found that there are many facing the same issue on redmi note 3 below are few of them trey the solution given by the useres it may work!
https://www.reddit.com/r/LineageOS/comments/6dlael/all_may_2017_updates_get_stuck_on_boot_screen_in/
https://www.reddit.com/r/LineageOS/comments/6skla4/last_few_months_of_lineageos_stuck_on_boot/
https://forum.xda-developers.com/redmi-note-3/how-to/recent-months-lineageos-stuck-boot-t3653960
Thank you Anvith. I will try the suggestions and will let you know. Meanwhile it would be great if you could suggest me any way to revert back to MIUI, just in case nothing works. Please note that I haven't kept any backup of original ROM.
Thanks
Sujeet