get the child categories, using get_categories();
then loop through them with a foreach loop, using WP_Query() :
<?php $cats = get_categories('child_of=".get_query_var("cat'));
foreach ($cats as $cat) :
$args = array(
'posts_per_page' => 3, // max number of post per category
'category__in' => array($cat->term_id)
);
$my_query = new WP_Query($args);
if ($my_query->have_posts()) :
echo '<h3>'.$cat->name.'</h3>';
while ($my_query->have_posts()) : $my_query->the_post(); ?>
<?php /*general loop output; for instance: */ ?>
<a href="https://wordpress.stackexchange.com/questions/17268/<?php the_permalink() ?>"><?php the_title(); ?></a> <br />
<?php endwhile; ?>
<?php else :
echo 'No Posts for '.$cat->name;
endif;
endforeach; ?>