Lead Developer
Hmm, without knowing the code, your previous theme most likely either:
a) Showed the full posts on the blog (no excerpt, stripping elements) – you can set your blog to do this in the Customizer.
b) Used code like this to pick out the first image in the post and applied it as the featured image: http://wordpress.stackexchange.com/questions/131411/programmatically-set-first-image-as-featured
I’m not sure of another way to do it unfortunately.
Using the code, there’s probably a way to look for images with a specific class, or ignore images with a specific class, but that would require some deeper digging.