I first read this link (http://website-experts.ca/how-to-fix-showing-latest-tweets-on-your-website/) before I came here, but I think the problem is mainly with the Plugins. I have a ton of sites on Rackspace (shared cloud sites) and do a bunch of custom work with Genesis Child Themes. I just set one up a site and was trying to get the Genesis Latest Tweets Widget (plugin) to work. I even contacted Rackspace support and we worked on it for 30+ min. I installed another plugin to see if I could get that to work - it didn’t either. It may have something to do with the Trust Level of the shared servers your host provide, which would mean the plugin developers need to do something about it.
That being said, … there is good news.
In your twitter account go to https://twitter.com/settings/widgets/ and create a Widget for your website - after you save the widget you create a snippet of code that you can past into a Text Widget for your sidebar (or other location). That seem to work much better for me than a plugin, though I just instituted it so I haven’t thoroughly tested it.
Maybe it will help you guys.