Update: Way better answer.
This is now a feature in the beta instance of Canvas (and should be live soon). Here are the notes: https://community.canvaslms.com/docs/DOC-8024#jive_content_id_Course_Settings
---Original Post-----
Matt,
You can change the course card background to an image with the css attribute content. Here is an example:
#DashboardCard_Container > div > div:nth-child(3) > div.ic-DashboardCard__header > div.ic-DashboardCard__header_hero {
content:url('path/to/image');
}
If you want to apply a different image to each course, you will need to use javascript to apply the content style with the appropriate url.
Hope this helps,
Alex