#Start the MySQL server instance or daemon with the --skip-grant-tables option (security setting).
$mysqld_safe --skip-grant-tables

$ mysql -u root mysql
$mysql> UPDATE user SET Password=PASSWORD('my_password') where USER='root';
$mysql> FLUSH PRIVILEGES;

#If you face the unknown field Password error above use:
#update user set authentication_string=password('my_password') where user='root';
#Finally, restart the instance/daemon without the --skip-grant-tables option.

$ /etc/init.d/mysql restart
#You should now be able to connect with your new password.

$ mysql -u root -p