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> 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

  1. Stop mysql
    systemctl stop mysqld
  2. Unset the mySQL envitroment option so it starts normally next time
    systemctl unset-environment MYSQLD_OPTS
  3. Start mysql normally:
    systemctl start mysqld

Try to login using your new password:

  1. mysql -u root -p