I had the same problem with my Sony Xperia Pro (MK16i), running on Android 4.0.4 (ICS).
Whenever I install an app that turns the screen off on a shake, twist or turn the problem occurs: multiple copies of an SMS.
When I disable it or better even, uninstall it, everything turns out normal.
Apparently the App prevents SMS-Receive-Reports to be sent back tot the SMS provider (phone-company) 's Servers.
Then these think the SMS was not received (phone off or something alike) ant try is after 10 minutes or so (depending on the provider) again.
This causes multiple receipts. Until the phone get the opportunity to send a SMS-Receive-Reports back. Then the whole circus stops. Until the next SMS.
Try to see if something uses the Device-control slots (see settings-security). Try to disable it or uninstall it.
Or get a newer phone or Android version.
Otherwise, I also don't know.
Cheers,
Imre