I'll say first that the suggestions involving servers are probably the best route, followed by purpose-built non-free solutions (like Acronis). Failing those, on the free side of things I have used Cobian Backup before with some success (not their latest version, I was probably on 6 or 7 at the time). It's a simple, free, backup program that creates full or incremental backups of selected files/folders on a schedule. One interesting twist of the program (when I used it) is that it did not compress or encrypt data, it just backed it up. This can actually be an advantage in some organizations as the staff may not have the ability to figure out how to open an archive.

I would imagine that to use it in this way you would determine the best settings, then copy them to each computer. On the backup server you would create a folder for each client and point each client at its network folder.