I found a backup of my iPhone which was done yesterday. Next I found a brilliant utility named iPhone Backup Extractor. I'll tell you what — normally I do not buy paid versions of software if the free version is doing what I can. This app doesn't have a paid version, so I just went and donated €5.00 to author — it's a brilliant piece of technology, dead simple and works. All what it does is just extracts the data from the backup file(s) (there're zillions of them in iPhone Backup directory) in form they can be placed to the iPhone. Well - that's what I did. PhoneView allowed to copy and paste these files where they belong — and the next time I fired up my "Messages" application, I was pleasantly surprised by each and every SMS since I've purchased my iPhone.
Dear Mr Padraig Kennedy — you have saved my day. And I appreciate it. My best regards to you and to what you're doing.