So to reset the root password, you still start mySQL with --skip-grant-tables options and update the user table, but how you do it has changed.
1. Stop mysql:
systemctl stop mysqld
2. Set the mySQL environment option
systemctl set-environment MYSQLD_OPTS="--skip-grant-tables"
3. Start mysql usig the options you just set
systemctl start mysqld
4. Login as root
mysql -u root
5. Update the root user password with these mysql commands
mysql> UPDATE mysql.user SET authentication_string = PASSWORD('MyNewPassword')
-> WHERE User = 'root' AND Host = 'localhost';
mysql> FLUSH PRIVILEGES;
mysql> quit
*** Edit ***
As mentioned my shokulei in the comments, for 5.7.6 and later, you should use
mysql> ALTER USER 'root'@'localhost' IDENTIFIED BY 'MyNewPass';
Or you'll get a warning
6. Stop mysql
systemctl stop mysqld
7. Unset the mySQL envitroment option so it starts normally next time
systemctl unset-environment MYSQLD_OPTS
8. Start mysql normally:
systemctl start mysqld
Try to login using your new password:
7. mysql -u root -p
