From what I understand I can tell Google to remove pages from their archive if I add the header:
<meta name="ROBOTS" contents="NOARCHIVE" />
But won't this also remove the page from search results? I only want the "Cached" link removed.
No, it will not remove your site from the listings, only from the archive. So your NOARCHIVE idea is correct. You can read more about how Google treats all of this on Google's own blog: http://googleblog.blogspot.com/2007/02/robots-exclusion-protocol.html