I would say most, if not all yeshivas of today learn in general (and specifically the study of Gemara -- and unfortunately ONLY Gemara with little mussar, little halacha and probably no Tanach) in the way that R' Chaim of Volozhin prescribed. That most would call "iyun" or his type of "iyun". Many Yeshivas followed in this way: Slobodka, Mir, Ponevezh, Kelm, Telz, etc. Until today the similar style of limud continues in all Yeshivas (probably both litvish and chassidish.)
However to me this is quite troubling, what gave him the authority then to change the way people should learn? More importantly what gives the chachamim of our generation to continue learning in a way that goes against the gemara in Avodah Zara Daf 19 that a person should learn a lot first, then do more "iyun" later. Going against Rambam in Hilchos Talmud Torah, perhaps the Shulchan Aruch and nosei Kelim itself and for sure the Shulchan Aruch HaRav in Hilchos Talmud Torah (which is built upon what the Tur, Rambam, Chazal and others said.) Which is to learn in a way of "bekius", until one knows everything well, then later in life to learn "iyun". He even prescribes how each type of person should learn (i.e. one who can understand well, one who doesn't understand so well, etc.) Although perhaps the Litvish community wouldn't give so much thought to this I'll note as well that The Arizal, Sha-ar Ha-Mitzvot Va-etchanan, page 79 right column, writes that iyun should be only be learned an hour or two a day (and that this itself was only for the "keen minded", but those who have "dull" brains should not even devote that time to iyun.)
Perhaps my biggest question on how this started was the fact that R' Chaim of Volozhin was a talmid of the Gra who himself writes in Mishlei Chapter 6 Verse 8 (starts at the bottom of the page that I linked to) that a person should spend the first part of his life learning a lot (bekius) and then later in life spend time more with "iyun". How could he go against his own Rebbe?
How can the generations ignore what Chazal and the great Chachamim that came before us prescribed. Not to much that the iyun often or never brings someone to halacha l'maseh (practical Jewish Law) which is perhaps the most important thing.
My question is a big one, however if we'll say that this whole way of learning started with R' Chaim then I have a kasha on him, how could he go against his Rebbe, the Gra.