sudo mysql -u root
mysql -u root
ERROR 1698 (28000): Access denied for user 'root'@'localhost'
The solution is to provide a password for the root mysql account (if you've not done so already). The error message you're receiving is because a password is required, and you have not provided it.
$ mysqladmin -u root password 'newpassword'
or if you'd already set a root password (which I doubt, otherwise you wouldn't be able to log in via sudo) then it would be
$ mysqladmin -u root -p'oldpassword' password 'newpassword'
Mysql users are not linked with unix users, unlike postgres.