In the only really big Asp.Net MVC application I did so far, I was required not to use anything using the Session.
As TempData uses session, I tried figuring out TempData CookieProvider (was like a year back though) and just could not get it to work in our project setup. (Others also tried and failed).
So we went and built the whole system avoiding TempData completely.

I'm moving on to other projects now, and seems the above project gave me an aversion to using TempData.
From my point, I can see it being useful messages (feedback) to the View at best.
Though I did not look into Azure, I do have a project coming up that will end up in the cloud, and not even sure how well session dependent functionality will work on that.

Is it just me? I've seen some code that put full models into TempData and then redirect to another action method which is pretty scare (imo).