I've read his two volumes on the Mediterranean in the 16th century, as well as his three volume "Civilization & Capitalism" series. I find his writing to be enthralling, and he analyzes economic and material life in a way that I haven't found anywhere else. I realize that his work is somewhat outdated at this point (and extremely eurocentric at that). I realize that many of his works are predicated upon the possession of a Eurocentric or Western perspective, but I would be very interested in seeing if there are any current historians who do similar things from a non-Western one.
Braudel was really part of the early generation (second) of the Annales school, which in later years - Emmanuel le Roy Ladurie onwards - began to turn towards culture (mentalités) rather than so much emphasis on the prior material concerns as geography and economy. So you're right, in short, that there won't be much that tries to cover the same period in the same style. That said, I recommend trying Le Roy Ladurie's Montaillou, or even try a historiographical review for crucial discussions on annales methodology, from weaknesses to strengths, and how the approach itself evolved - e.g. Burke's The French Historical Revolution: The Annales School (1990) or Eric Dursteler's biographical entry on Braudel in Daileader & Whalen's French Historians (eds., 2010).
But since you're especially interested in non-Western views, see the work of Sanjay Subrahmanyam. As I mentioned in a previous post, I have the utmost respect for him:
[He's] an Indian-born historian specialising in the early modern era, who formerly held an endowed chair at UCLA before now taking up a place at the formidable Collège de France in Paris, and who's an absolute giant in the field of global history. He happens to be a polyglot historian, one of the very few who's capable of working in a whole multitude of international languages, enabling him to synthesise secondary scholarship and of course have direct access to the sources -- and these include fluent reading mastery of 11 different languages: "Persian, Urdu, Telugu, Tamil, Portuguese, English, Spanish, German, French, Italian and Dutch", to quote from his Books&Ideas interview (see http://www.booksandideas.net/History-Speaks-Many-Languages.html).
There are a few videos of his talks on YouTube, and if you can understand French, see particularly his inaugural for the Collège de France, in which he discusses Braudel. If not, those comments can be found in the introductions to his different works. I should note that his books aren't limited to the Mediterranean but adopt a global outlook, so be aware of that if you're looking for work on the Mediterranean. On the other hand, he explicitly draws much inspiration from Braudel, while noting criticisms of him in - as you point out yourself - the relative eurocentrism of his discussion. If global interconnectivity interests you and you wish to pursue it after the early modern period (where Subrahmanyam mostly focuses), then see also Chris Bayly's Birth of the Modern World. Of course, with Global History on the ascendent ever since the global turn about almost two decades ago now, there are many more I can suggest if you'd like.