Disable "Blogroll" or "WordPress Dashboard News" section in WordPress v.4.1?

The 'WordPress News' widget is dashboard_primary and (likely for reasons buried somewhere deep in core) sometimes it reappears when using

add_action('wp_dashboard_setup', 'remove_dashboard_widgets');

to remove widgets... often can instead use

add_action('admin_init', 'remove_dashboard_widgets');

so, if having issues with

function remove_dashboard_widgets () {
    remove_meta_box('dashboard_primary','dashboard','side');
}
add_action('wp_dashboard_setup', 'remove_dashboard_widgets');

...instead try like this

function remove_dashboard_widgets () {
    remove_meta_box('dashboard_primary','dashboard','side');
}
add_action('admin_init', 'remove_dashboard_widgets');

and, fwiw, for a network dashboard try using

function remove_network_dashboard_widgets() {
    remove_meta_box('dashboard_primary', 'dashboard-network', 'side');
}
add_action( 'wp_network_dashboard_setup', 'remove_network_dashboard_widgets' );

You can hide them using the screen options or add this code to your child themes functions file:

add_action('wp_dashboard_setup', 'remove_dashboard_widgets');

function remove_dashboard_widgets () {

      remove_meta_box( 'dashboard_primary', 'dashboard', 'side' );      
      remove_meta_box( 'dashboard_secondary', 'dashboard', 'side' );      


}

Source: wp_dashboard_setup

If you want to remove metaboxes from the dashboard page you can add this to functions.php

function remove_dashboard_widgets () {

  remove_meta_box('dashboard_quick_press','dashboard','side'); //Quick Press widget
  remove_meta_box('dashboard_recent_drafts','dashboard','side'); //Recent Drafts
  remove_meta_box('dashboard_primary','dashboard','side'); //WordPress.com Blog
  remove_meta_box('dashboard_secondary','dashboard','side'); //Other WordPress News
  remove_meta_box('dashboard_incoming_links','dashboard','normal'); //Incoming Links
  remove_meta_box('dashboard_plugins','dashboard','normal'); //Plugins
  remove_meta_box('dashboard_right_now','dashboard', 'normal'); //Right Now
  remove_meta_box('rg_forms_dashboard','dashboard','normal'); //Gravity Forms
  remove_meta_box('dashboard_recent_comments','dashboard','normal'); //Recent Comments
  remove_meta_box('icl_dashboard_widget','dashboard','normal'); //Multi Language Plugin
  remove_meta_box('dashboard_activity','dashboard', 'normal'); //Activity
  remove_action('welcome_panel','wp_welcome_panel');

}

add_action('wp_dashboard_setup', 'remove_dashboard_widgets');

Just comment out the ones you want to keep. The first value passed is the metabox ID, so you could also remove other metaboxes that are added by plugins. Just inspect the metabox with your browser and grab the metabox ID.