I found a lot of articles on the internet telling me that I should do a Tools / Deactivate. But, because I changed a system and I changed a serial number, I could not do that. Deactivate option was not available because all the time CS3 asked me to re-activate. And, I just wanted to start activation from beginning, with Activation Type: normal, not: Activation Type: Repair.
I did not found the solution right away, and that is the reason I am writing this post. So, just delete two files, and when you start some CS3 application, it will ask you for a serial number in the same way it asked you the first time. The two files that should be deleted are:
c:\Program Files\Common Files\Adobe\Adobe PCD\cache\cache.db
c:\ProgramData\FLEXnet\adobe_00080000_tsf.data
This is for Microsoft Windows Vista, but I am sure that you will manage to find these files on Windows XP.