Removing the dashboard:

/* Remove the "Dashboard" from the admin menu for non-admin users */
function wpse54085_remove_dashboard () {
    global $current_user, $menu, $submenu;
    get_currentuserinfo();

    if( ! in_array( \'administrator\', $current_user->roles ) ) {
        reset( $menu );
        $page = key( $menu );
        while( ( __( \'Dashboard\' ) != $menu[$page][0] ) && next( $menu ) ) {
            $page = key( $menu );
        }
        if( __( \'Dashboard\' ) == $menu[$page][0] ) {
            unset( $menu[$page] );
        }
        reset($menu);
        $page = key($menu);
        while ( ! $current_user->has_cap( $menu[$page][1] ) && next( $menu ) ) {
            $page = key( $menu );
        }
        if ( preg_match( \'#wp-admin/?(index.php)?$#\', $_SERVER[\'REQUEST_URI\'] ) && ( \'index.php\' != $menu[$page][2] ) ) {
            wp_redirect( get_option( \'siteurl\' ) . \'/wp-admin/post-new.php\');
        }
    }
}
add_action( \'admin_menu\', \'wpse54085_remove_dashboard\' );

Removing further parts of the admin menu:

function wpse54085_admin_menu_fixes() {
    global $current_user;

    if( ! in_array( \'administrator\', $current_user->roles ) ) {
        remove_menu_page(\'link-manager.php\');
        /* add more menu pages, depending on what you want to remove */
    }
}
add_action( \'admin_menu\', \'wpse54085_admin_menu_fixes\' );