Java apps have gotten a lot easier to run on Azure this week. Here's
a video on developing a simple app with Eclipse and deploying in
Tomcat on Azure:
http://microsoftpdc.com/Sessions/SVC50
I don't believe Java on Azure has the same restrictions as Java on
GAE. For example, you can write to the local filesystem and use
threads. Using the REST-based Azure Tables datastore (non-RDBMS) with
Lift's ORM might be tricky, but SQL Azure is available via JDBC. Or,
as far as I know Derby should work.
I think you need to be on Windows to develop though, even though the
example in the video just uses Eclipse. For one thing, the desktop
version of Azure is Windows-only.