[RndTbl] My MySQL battle

OK, I've come to the conclusion that I'm doing something wrong, but can't for the life of me figure out what it is. Going over and over and over documentation, I must be interpreting something wrong, because I think I'm doing what it says.

I'm setting up a new box with CentOS. Installed MySQL, and it's running. Now I get to setting the password. Here's the issue.

I start with:

mysqladmin -u root password SQLROOTPASSWD.

No problem

Then I try:

mysqladmin -h myserver.mydomain.com -u root password SQLROOTPASSWD.

This is where things go South. I get the response

mysqladmin: connect to server at '(my server)' failed

"Access denied for user 'root'@'192.168.1.160' (using password: NO)'

I'm at a loss. Hours of fighting with this, and I'm nowhere... Help?