The issues were all "Web Sites" specific. I'm sure we could spin up machines and get everything to run - but that's much less enticing compared to AppHarbor / Beanstalk.
One example - if you have any processing in the application_startup, Web Sites won't work for you. Applications are randomly killed and restarted, and don't allow for a startup time, meaning outages when this happens. Customer support suggested using a fail over to work around this, but the fail over only kicks in once your app has been down for a certain amount of time - so you'll still have down time.
What if your running app relies on the processing done during startup?
Sure - we could re-architecture the app so that isn't a requirement, but then I'm making design decisions based on Azure constraints. Why would I when there are alternatives like AppHarbor?
Would you care to share any specifics? I'm curious to know what you're trying to do that doesn't work on Azure.