Yeah, newer cases, especially the ones you get from companies like Dell are made with thinner gauge sheets of aluminum and it's really a case (no pun intended) of "they don't make them like they used to".
If you look on the inside of the old hard drive caddies, there should be some rubber grommets where the drives are held in by. You can try to replicate that in the new case with some rubber washers to see if that helps. The goal is to put something (preferably rubber) between the hard drive and the case to dampen the vibrations. If that's not possible and the noise is really bothering you, there are dampening mounts you can buy that lets you install the hard drive into a 5.25" drive bays.
Something like this
http://www.performance-pcs.com/mitro...dampeners.html