As others have said, this is a very common method (“Mahasi approach”). It is taught by Mahasi Sayadaw’s meditation centres in Burma, and his students such as U Pandita and Chanmayay Sayadaw. The approach became very popular in Sri Lanka (probably in the 50’s and 60’s) and Thailand in the 80s.
It is the basis for much (but not all) of what is marketed as secular “Vipassana” or “Insight Meditation” in the West. The U Ba Khin/Goenka approach is another Burmese approach. That is rather different.
Other teachers, such as Ajahns Brahm, Sujato, Pa Auk Sayadaw, etc, teach a more absorbed samadhi approach.
There are a huge variety of meditation approaches in Asia. Only a few have become popular in the West.
To get some idea of the variety, I recommend James Baraz’s has a series of talks that are based on Jack Kornfield’s book Living Dharma (formerly LIving Budddhist Masters):
The book covers 12 modern meditation teachers from Burma and Thailand (most of whom are not longer living, hence the title change…),
In his talks: http://dharmaseed.org/teacher/86/?search=buddhist+master+series, Baraz adds some of his own experience with these teachers, as do some of the audience. He also gives some short guided meditations on each approach.
These talks (or the book) are worthwhile in dispelling the notion that there are only one or two “normal” meditation practices. Here we have Sunlun Sayadaw’s heavy breathing (and heroic sitting), Ajahn Naeb’s attention to postures as a route to understanding dukkha, and teachers who focus on body, feelings, mind, etc…
And though he’s not in Kornfield’s book, a later talk from Baraz on U Tejaniya http://dharmaseed.org/teacher/86/?search=tejaniya is in the same vein.
As far as what is “best”, I agree with Patrick Kearney, whose retreat talks Audio | Dharma Salon usually contain something also these lines:
So when we talk about method we are talking about the systematic training of attention. If I’m going to pay attention I have to have some method. And ther are multiple methods available. We quickly worked out five just for looking to see who is in the room. [This refers to an earlier discussion of possible ways of scanning the room.]
And these are the various meditation techniques in the spiritual supermarket. And I think it’s important to understand that there isn’t one true technique, any more than there is any such thing as the one true washing powder. It doesn’t exist, but the marketeers will tell you that only one washes whiter than white, and you get exactly the same phenonenon in the spiritual world: “Oh yes, meditation is very good for you but only this one will get you enlightened.”
As meditators it is important to understand that this is completely nonsense [Patrick, being Australian, uses stronger language…]. The job of the pracitioner is not to discover the one true technique. It’s your job to discover the method that works best for you. And it’s not necessarily the method that works best for the teacher. What the teacher is teaching is usually what worked for him or her. But it’s not necessarily what works for you.