Your best best for the moment is to use a standard WebDAV client.
(There are many available for all platforms!)
Desktop and Mobile clients "are coming".
RE "Dropbox-like" -- Well yes sort of and sort of not. CozyCloud is much more than just "File Storage" (that's just one aspect).
Think of CozyCloud like your own "Private Cloud" with "Private Data", "Private Apps" and "Private Sharing".
cheers
James