Wrong priviledges after moving

I’ve successfully moved my WordPress site to a new hosting server (the URL remains the same), and everything is fine, except one thing – the default prviledges for folders and files (755 and 644) aren’t sufficient for WordPress to be able to write to folders and files anymore. In order to make them writable, I have to change every folder into 777 and every file into 666. So it seems that the server doesn’t know that Worpress is the “owner”. Does anyone know where the problem is?

The new server is a privately owned on-line Linux server, it’s not a regular shared hosting of a commercial hosting provider. The owner is willing to help resolve this, but needs a hint where to search for a solution.