With Linux, this is a lot more difficult and requires more third party add-ons.
Only allow root to mount disks. Your users shouldn't have access to sudo, su, or the root login, anyway. Pretty simple, really; locate the mount binary for your system (/bin/mount is a good bet; if your mount binary resides elsewhere, you'll have to modify the commands below to reflect that), then do the following:
chown root:root
chmod 0750
Done. Now, only root can even execute the mount binary, so only root can mount disks, and that will include flash drives.
It does get a little more complicated if you need to be able to mount network shares, but you should be able to add those to
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