In the functions.php you will find a function that returns an array of related posts of the currenty post. It uses the categories of the current post to collect all posts that have the as much same categories set as possible.
Note: This German blog post explains how the idea was born.
Another note: I'm not working with Wordpress anymore so I won't develop this function further. But if you spot any mistakes, feel free fix it and send a PR.
This is just a function and does not need any installation. Just copy and paste this function into your theme's functions.php file.
Within the loop call the function rsRelatedPosts($count, $sort) where $count is the amount of related posts you want to have and $sort is how the post should be selected.
$sort has to be one of the orderby-parameters of WP_Query (See Wordpress Codex)