There's not yet an official interface or UI for doing so like in the old index.
However, there's an 'unofficial' way to do this as the software supports access to the original resource using the replay url form.
{"urlkey": "org,commoncrawl)/", "timestamp": "20150302032705", "url": "http://commoncrawl.org/", "length": "2526", "filename": "common-crawl/crawl-data/CC-MAIN-2015-11/segments/1424936462700.28/warc/CC-MAIN-20150226074102-00159-ip-10-28-5-156.ec2.internal.warc.gz", "digest": "QE4UUUWUJWEZBBK6PUG3CHFAGEKDMDBZ", "offset": "53235662"}
You can access the original resource via this url, using curl or wget:
curl http://index.commoncrawl.org/CC-MAIN-2015-11/20150302032705id_/http://commoncrawl.org/
wget http://index.commoncrawl.org/CC-MAIN-2015-11/20150302032705id_/http://commoncrawl.org/
Note the format here is: /CC-MAIN-2015-11/ + the timestamp + id_ + / url
Please note that this capability is part of the pywb replay software, and may change in the future for CommonCrawl. It's not guaranteed to work in all cases..
This replay serves the original response http headers as well, which may not be consistent with content and may not always work in the browser.