Starting from the CM12, CyanogenMOD 12 which is Android 5.0 Lollipop, it becomes noticeable that you can backup SMS using the native (Stock) CM SMS application and restore it at a later time.
And because the SMS is backed up in .xml format, you can easily restore it on either the same phone or another using any other SMS management app that supports (can decompress) .xml file types.
How to Backup SMS Using CM (CyanogenMOD) SMS Application
Step #1: your Android device must be running on CM12 (CyanogenMOD 12) or higher.
Step #2: Open CM12 SMS application and tap Option and go to Settings .
Step #3: Click Impoort and Export and then, select either Backup or Restore depending on which you intend to do.
If you're restoring the SMS Backed up with CM Native SMS application using another application, you should first unzip the Backed up files and then, rename the one ending with SMS and add .xml as it suffixes so that it will look like: 20160804_SMS.xml
How to Locate the Backed Up SMS Folder
Step #1: Open your file manager or download and install a file manager such as ES File Explorer.
Step #2: from the file manager, located BackupSms and open it. This is where all the SMS backed up using the CM Native SMS application are saved.