SMS Backup + 1.4.2
the author of this app says:
Automatically backup SMS, MMS and call log entries using a separate label in Gmail / Google Calendar. It is also possible to restore SMS and call log entries back to the phone (MMS not supported yet).
Remember to enable IMAP in Gmail and set the correct time/zone.
If you get a "You do not have permission to..." error or other problems during the first connect check the website for workarounds.
Please note that auto backup might not work reliably when SMS Backup+ is installed on the SD card. Future version will hopefully fix this, in the meantime please move it back to phone memory if possible.
For those of you worried about app permissions: there is a detailed explanation in the FAQ section of the website.
A few people have asked about a donate option - there is one (via PayPal), it's hidden in the "About" screen.
Also, please consult the FAQ on the website before sending me an email.