Recently the Android app I had been using to backup my SMS/MMS messages became unsupported, so I looked around for an alternative.
At the moment, it looks like SMS Backup+ is the best tool for the job.
It will backup SMS, MMS attachments, and the call log to Gmail and assign a label to them. It also has a range of other settings, like syncing the call log with Google Calendar.
And, on top of that, the app is free and ad-free (I did send a $5 donation as the app was easily worth that much to me).