Wordpress loop, most recent post from multiple category in custom category order?

I have a site with posts in 16 categories ("departments"). The home page is a grid of the current (newest) post in each category.

I would like to create the HTML for displaying a department's current post and use a word press loop to repeat this 16 times to get the grid.

ALSO - and her is the part I need help with - the client wants to specify the categories AND the order they appear.

So, my page is the newest post from "Dining", then the newest from "travel", etc.

Here's the site currently where the previous designer did this by hand in a table every month, which is silly.

via Chebli Mohamed