First lets understand why this change has been made. Facebook had this to say about the topic.
So it's all about fake news. Although I understand Facebook's position on this, I do not agree completely with this change. Lets look at what you can do now to control which image appears as a link preview thumbnail...
As Facebook said, the Open Graph meta tags are the currently supported way to modify link previews. Fortunately this is easy to add to a CMS or Blog system. It only took me a couple of hours to have a full end-to-end solution in my Blog's code working.
The trick is to implement tags like this in the <header> of your generated HTML page...
The og:image tag is the one that is required, others are optional but I would recommend using them. This works the same way as back when the 'picture' property was supported with the API. Simply put the URL of the image into the content attribute and you're done. The image here needs to meet some guidelines e.g. it has be be less than 8Mb in size for Facebook to download it.
The og:image:type tag specifies the MIME type of the image. The og:image:width and og:image:height provide the width and height of the image (in pixels) respectively.
A PHP example on posting links is covered in Facebook's documentation here. Don't forget to use the URL Debugger after making your changes to ensure Facebook is behaving as you were expecting.