I am very biased and forgive Kay for most of his faults, but I'll give my thoughts on this. Under Heaven feels like Kay is writing 'about' characters rather than from their perspective. It doesn't feel as 'personal' as Lions of Al-Rassan or Song for Arbonne. It's hard to describe, admittedly. I do love the work, however. I like the descriptions of the nature in the work and feel that Kay's commentary on historical forces works best in this work.
River of Stars is interesting in that it explores the eventual consequences of the political conflict in Under Heaven. It makes the books feel like a duology of sorts. I also appreciate that it has a character who is thinking of his role in history. The book is very much focused on military events. In all honesty, it sort of feels like Kay tried to do his version of a Bernard Cornwell novel.
SeanF, I never made the association between Suleiman and Gurcu. Thank you for pointing that out to me. I always found myself thinking of Suleiman when I read about Brandin in Tigana, however; Brandin and Dianora feel like a parallel to Suleiman and Roxelana to me. In fact, I sort of see Tigana as a hypothetical scenario in which the Ottoman Empire (as symbolized by Ygrath) manages to become a dominant power in the Italian peninsula. Not quite sure what this makes Alberico, admittedly. Maybe some amalgamation of French/Spanish/Norman influence?