I actually found out how to back up SMS and MMS (along with the multimedia attachments) by accident when trying to troubleshoot another issue. The issue I was having was deleted MMS message threads would come back blank with a date of December 31, 1969. Without going into details, the only way to solve this was to delete the database that housed the SMS and MMS data because it was corrupted.
Using the app History Eraser did not help, nor did clearing the app data/cache, nor cache paration and Dalvik Cache helped with the issue. History Eraser gives you the ability to remove ALL messages (gives you a clean slate), but the problem comes back when I receive an MMS and decide to delete it. It comes back as a blank message with the 1969 date.
Time travel jokes aside, threads from different Android forums came to a dead end. All were dancing around the actual fix, so after reading them all, I got a better understanding of how Android stores it's SMS and MMS data. Know this helped fixed the problem, but also gave me a better way to backup and restore my messages along with the attachments. It's a boon to anyone who is ROM flashing addict.
Without further ado, here's where the magic goes:
In the databases folder, the file "mmssms.db" holds your SMS and MMS text messages, while the app_parts folder contain all of the MMS's attached files (photos, videos, audio, etc).
If you have the Dec 31, 1969 issue like I did, and you don't care to back up your messages, you can delete the "mmssms.db" file, as well as clear the contents out of the app_parts folder. You can then clear your Messenger app's data and cache and reboot your phone. You will have a clean slate. This is because the database file was corrupted and utilities that delete messages just deletes the contents inside of the bad databse file instead of deleting the actual file itself.
After all of this, your Messenger app will recreate the missing "mmssms.db" file from scratch.
Now that you know how to fix the issue, as you can imagine, backing up the same file along with the app_parts folder will allow you to restore your message contents if you ever need to flash a new ROM or move to a different phone.
All of this requires root, of course. You have many options to access these files, but the easiest for me is to use a root file manager such as Root Explorer to navigate to the path and copy & paste the files and folders to my external SD card. When restoring, you may want to take note of the permissions of the original files and folders so you can set them back on the restored copies.
The funny thing is I believe the corrupted database file was caused by using an SMS backup app. It saves your SMS and MMS in an XML file and restores the contents back into the database. The issue may be that MMS backups are not well supported, (they contain attachments and treated like email, unlike SMS). Unfortunately, it's still the best way to back up and restore individual messages with a surgeon's precision. Just make sure not to backup any MMS messages!