If you move your datadir, you not only need to give the new datadir permissions, but you need to ensure all parent directories have permission.
I moved my datadir to a hard drive, mounted in Ubuntu as:
/media/*user*/Data/
and my datadir was Databases.
I had to set permissions to 771 to each of the media, user and Data directories:
sudo chmod 771 *DIR*
If this does not work, another way you can get mysql to work is to change user in /etc/mysql/my.cnf to root; though there are no doubt some issues with doing that from a security perspective.