Show posts organized by Author

<?php
//displays all users (sorted by display_name) that are post authors with their avatar and their posts (titles)
$blogusers = $wpdb->get_results( "SELECT ID, user_nicename from $wpdb->users WHERE 1=1 ORDER BY display_name" );
if ($blogusers) {
  foreach ($blogusers as $bloguser) {
    $user = get_userdata($bloguser->ID);
    $args=array(
      'author' => $user->ID,
      'post_type' => 'post',
      'post_status' => 'publish',
      'posts_per_page' => -1,
      'caller_get_posts'=> 1
    );
    $my_query = null;
    $my_query = new WP_Query($args);
    if( $my_query->have_posts() ) {
      echo '<p>User ID ' . $user->ID . ' ' . $user->user_firstname . ' ' . $user->user_lastname . '</p>';
      echo get_avatar( $user->ID, 46 );
      while ($my_query->have_posts()) : $my_query->the_post(); ?>
        <p><a href="<?php the_permalink() ?>" rel="bookmark" title="Permanent Link to <?php the_title_attribute(); ?>"><?php the_title(); ?></a></p>
        <?php
      endwhile;
    }
  wp_reset_query();
  }
}
?>

Michael! This was amazing, I just have to tweak it a little but thank you! Will let you know of the final results.

Once again you have saved the day.