Office 365 boasts a new feature titled Skydrive pro. It's really just Sharepoint workspace libraries re-branded and modified but it is more user friendly, and somewhat resembles Dropbox. One catch is users are given a 7GB limit on their individual Skydrives which cannot be increased, so a document is provided to offer workarounds. This might also be a good reason to keep those file servers going at your on-premise locations.
