While the Sprint Burndown is no longer required per se, monitoring sprint progress in some fashion is still required, and it is still required to sum the amount of work remaining in the sprint on a daily basis.
Most teams use a sprint burndown for this purpose, and while you don't have to use a sprint burndown, you do *have to use something.*
What units you use is flexible. Some teams use ideal hours, real hours, ideal days, story points, the "number of tasks", number of acceptance tests, etc. IME, most teams use something like ideal hours or ideal days, with the more advanced ones doing "micro-stories" and burning down in story points.
More about micro stories here: http://www.scrumcrazy.com/User+Stor...ro+Stories