Why are Jhanas controversial? I have heard that some teachers do not teach the Jhanas, others do. Yuttadhammo mentions the great debate about Jhanas. What is the controversy about? Whether they are useful towards realizing Nibbana/Nirvana? Is the debate relevant to all Buddhist traditions, only some, or are traditions split up in their views about Jhanas?
The debate centres around the Theravada commentarial interpretation of the suttas along with the teachings in the Abhidhamma, Patisambhidamagga, and Visuddhimagga.

Basically, there are two schools of thought, one which subscribes to the above teachings and one that rejects them.

The teachings in question are those that describe two potential paths: 


jhānalābhi - one who obtains at least the first jhāna
sukkhavipassaka - one who only obtains momentary concentration.


For example, in the Visuddhimagga:


  Herein, “purification of view” is the correct seeing of mentality-materiality. One who wants to accomplish this, if, firstly, his vehicle is serenity, should
  emerge from any fine-material or immaterial jhāna, except the base consisting of
  neither perception nor non-perception, and he should discern, according to characteristic, function, etc., the jhāna factors consisting of applied thought,
  etc., and the states associated with them, [that is, feeling, perception, and so on]. 
  
  ...
  
  But one whose vehicle is pure insight, or that same aforesaid one whose vehicle
  is serenity, discerns the four elements in brief or in detail in one of the various ways given in the chapter on the definition of the four elements (XI.27ff.).
  
  (Path of Purification, pp. 609-10)


Those who reject this split point to various teachings to support the idea that the samatha jhānas are necessary for attainment of the path.

An article that looks interesting from an academic point of view (I only read the abstract) is here:

http://www.tkwen.theravada-chinese.org/A%20Study%20of%20Sukkihavipassaka%20in%20Pali%20Buddhism_final.pdf

From the abstract:


  This thesis reveals that the concept of arahants who lack the first form-sphere jhāna is accepted not only by the Theravāda but also by the Sarvāstivāda, the *Satyasiddhisāstra, and the Yogacārabhūmiśāstra. Since various Buddhist schools in India unanimously advocate the idea that there are arahants who have not achieved the form-sphere jhāna, this research concludes that the dry-insight meditative approach and dry-insight arahants are not an invention by Theravādin commentators, but a common heritage which was most probably handed down from the time of the Buddha and then shared by various Buddhist schools.


I imagine reading through the thread I linked to in the other question would give you a good idea of the controversy. There was a heated debate many years back between a Burmese and a Sri Lankan monk on the topic that was highly academic, but still a fun read:

http://static.sirimangalo.org/mahasi/Satipatthana%20Vipassana:%20Criticisms%20and%20Replies.htm