Post Titles Listed by Category on Blog Index Page

I am creating a blog with the homepage to serve as the blog index page. I’d like to list the post title in date order by category, in groups of 3. You will see on the screenshot the categories listed on left, with a thumbnail, and the most recent 3 posts in that category listed to the right of them.

Screenshot

How would I do this?

Alternatively, is there a widget I could use, just making the homepage a static page and use the widget to populate posts as such?

Thanks.