After updating to WordPress 3.9 the Twitter and Meetup embed URL's do not display like they should. All that displays is the URL. When I switch to the Twenty Fourteen theme the embeds display like they are supposed to. This points to Atahualpa being the problem.
Link to the post with the problem
http://www.texbiker.net/blog/
Look at the bottom of the first post.

What can I do to fix the problem?
I am using Atahualapa 3.7.15.

I don't see any links. I just see the social sharing buttons.

you have a caching plugin active that could be part of the issue or masking it.

What version of Atahualpa?
Please provide a url to the widget download.

Version 3.7.15. I turned off the caching but the problem is still there.

I am using version 3.7.15. Look at this post at the bottom where it says "Tweets". I deleted the cache but the problem is still there.
http://www.texbiker.net/blog/2014/04...st-april-16th/

Look at the top post and scroll down to the section "Tweets". Below there are all of the URL's .

Where did you put those links and what is the code you used?

I put the URL's in the visual editor. The only code required is the URL. WordPress takes the URL and embeds the Tweet like you see on Twitter. Same for Meetup. I also tried the text editor but it did not solve the problem.

I used the visual editor and the text editor to paste the Tweet URL into the post.

Have you tried just using a standard hyperlink?