The best open source alternative to SMS Exporter is SMS Import / Export. If that doesn't work for you, our users have ranked six alternatives to SMS Exporter, but unfortunately only two of them is open source. If you can't find an alternative you can try to remove all filters.
SMS Import / Export imports and exports SMS and MMS messages, call logs, and contacts from and to (ND)JSON / ZIP files. (Contacts import and export are currently functional but considered experimental.) It does not require root.
Epistolaire is an Android app to backup SMSes AND MMSes (unlike most "SMS backup" apps), in a nutshell.
