RebelMouse allows the aggregation of Facebook, Twitter, Instagram, Google+, LinkedIn, Flickr, YouTube, Tumblr, and Pinterest accounts, as well as any specified RSS feed. The aggregated contents can then be fed straight to a page, or specified sections, with or without intervention.
If you wish to review, remove, edit, and add to each item the results can be fed to a Draft area before final posting. Items can be moved around your pages/sections after publication.
And then you can embed the whole thing into your website, like this: