Moving MySQL databases/folder to another disk
Yes, you can here is the steps to do it.
backup:

# service mysqld stop
# cd /var/lib/mysql
# tar -cvzf /tmp/mysql-backup.tar.gz ./*
# service mysqld start


restore:

# service mysqld stop
# cd /data/mysql
# tar -xvzf /tmp/mysql-backup.tar.gz
# service mysqld start


Notes: 


Cold backup will not work if you try to restore it in older versions of MySQL it will be better if you use the same version.
You need to make sure new directory is owned by MySQL user and it has the needed permission
New datadir is accurate inside my.cnf