Hi.
We recently finished migration from Exchange 2003 to Zimbra in 10 small offices worldwide, we're facing the same complaint but have solved half of it.
We have successfully been using the firefox drag & drop upload plugin for the upload part.
https://addons.mozilla.org/en-US/firefox/addon/2190
Probably it could be rewritten slightly to give fancier results, but for now it works.
However, this does not solve the download part. I think that what's being asked for should be achievable via a java applet. Look for example at the upload functionality in gallery2.sf.net.
I think this should be split into 2 issues though, 1 is download and then upload is another - very different things from the browser security model to tackle.
Best regards,
Maron