Query posts from an array of categories and their children

First, I’ve already checked the WP_Query Codex. I’ve already resigned myself to probably having to do a wpdb query, but here’s my problem:

1. If I query a single category ID, it returns all posts in that category and that category’s children.

Like this:

Top Level Category
	Post 1
	Post 2
	->Sub Category
		Post 1
		Post 2
	->Sub Category
		Post 1
		Post 2
		-> Sub Sub Category
			Post 1

2. If I query an array of category IDs, it returns all posts in those specific categories, but NOT their children.

Like so:

Top Level Category 1
	Post 1
	Post 2
Top Level Category 2
	Post 1
	Post 2
Top Level Category 3
	Post 1
	Post 2

I need a way to query the equivalent of the first example with several top-level categories and their children, but grouped separately in an array (e.g. array(group1, group2, group3))

Like so:

Top Level Category 1
	Post 1
	Post2
	->Sub Category
		Post 1
		Post 2
Top Level Category 2
	->Sub Category
		Post 1
	->Sub Category
		Post 1
Top Level Category 3
	Post 1
	->Sub Category
		Post 1

Any ideas?