I’m just developing a new site in Drupal using the Zen STARTERKIT and it’s been a while since I’d done a design without the “My Account” and “Logout” links being needed in the header.
To get rid of them, simply go to admin/structure/menu/settings and change or remove a menu in the Secondary Menus.
Simples
I'm a web developer based in the East Midlands, UK and if I keep up the current rate, I might have developed 3 million sites by the time I retire