MySmsBackup2 for Android
MySmsBackup2 for Android differs from MySmsBackup because the new version does not sync SMS's with the Gmail account, but instead it only sends the current snapshot of all Inbox and Sentbox SMS's to only one email sent to yourself on your Gmail accout, in the label "MySmsBackup" in reality an IMAP folder.
This is completely free and for now with no ads.
MySmsBackup2 for Android