It looks like the post_category is not used in the posts table anymore (if it ever did). instead you need to join with the post2cat table. The code below grabs 2 random posts from category 2.
<?php
global $wpdb;
$rand_posts = $wpdb->get_results("SELECT * FROM $wpdb->posts AS p, $wpdb->post2cat AS c WHERE p.ID = c.post_id AND c.category_id = 2 AND p.post_status = 'publish' ORDER BY RAND() LIMIT 2");
foreach ($rand_posts as $post) :
setup_postdata($post);
?>
