MySQL Multiple Joins in one query?

Just add another join:

SELECT dashboard_data.headline,
       dashboard_data.message,
       dashboard_messages.image_id,
       images.filename 
FROM dashboard_data 
    INNER JOIN dashboard_messages
            ON dashboard_message_id = dashboard_messages.id 
    INNER JOIN images
            ON dashboard_messages.image_id = images.image_id

You can simply add another join like this:

SELECT dashboard_data.headline, dashboard_data.message, dashboard_messages.image_id, images.filename
FROM dashboard_data 
    INNER JOIN dashboard_messages 
        ON dashboard_message_id = dashboard_messages.id
    INNER JOIN images
        ON dashboard_messages.image_id = images.image_id 

However be aware that, because it is an INNER JOIN, if you have a message without an image, the entire row will be skipped. If this is a possibility, you may want to do a LEFT OUTER JOIN which will return all your dashboard messages and an image_filename only if one exists (otherwise you'll get a null)

SELECT dashboard_data.headline, dashboard_data.message, dashboard_messages.image_id, images.filename
FROM dashboard_data 
    INNER JOIN dashboard_messages 
        ON dashboard_message_id = dashboard_messages.id
    LEFT OUTER JOIN images
        ON dashboard_messages.image_id = images.image_id