SharePoint 2007: Office Documents prompt for Authentication when trying to open from SharePoint site

But if you use Citrix or TS then its not this, but i suspect IE 6/7 , please check the authentication settings in the Security tab, and also please do not un-check client integration ( in CA site) as some of users suggested ( This is bad practice).
Cause:

This happens when Office detects that WebDAV is enabled on the server. If WebDAV is enabled then Microsoft Office also has to determine whether or not to open the document as ReadWrite or ReadOnly.  SharePoint acts as its own WebDAV Server.  Server 2008’s WebDAV Publishing role if enabled will interfere with SharePoint’s WebDAV functionality.