How do we veer away from Eurocentric ideas of history?

The definition of Eurocentric is something that is centered around or highlights the European culture and history. An example of Eurocentric is someone saying that european countries are better than others.

Just don't read books about europe.

Songbird John should note the Mongol Empire, rapidly established as the largest empire ever within a single land border.  Admittedly not did not last long.  

Also of note is the rapid spread of Islam in its early centuries, compared with the fairly slow spread of Christianity.

When you say “Eurocentric ideas of history”, what exactly do you mean?  I find this question incredibly vague.

There is a very good reason that kind of book is hard to find.  The problem is, most of the histories of non-European-based cultures are very static.  Sure, there were wars, and plagues, and genocides and migrations, but in the grand scheme of things, they were insignificant to the world we know today.  Other cultures were and are not innovators.  Everyone follows the lead of European or European descended culture.  The modern world as we know it is almost entirely the construction of European civilization, for better or worse.  All major inventions, the most momentous events, all are the work of Europeans and those descended from them.  Europeans connected the widespread cultures of the world through exploration and trade. When we think of 'civilization' we are thinking of European civilization.  What is the dominant clothing style around the world?  European, or Western.  What is the trade language of the world?  English.  Eurocentric history is THE history.  Without Europeans, and Euro-descended societies like the USA, the world would be a benighted, backward planet, and it would become so if Euro-culture ever fell. @John P the Mongols and the rise of Islam are two events initiated by non-Euro cultures that had lasting effects and are covered in standard histories. Note that these events are both significant largely due to their effects on and the reaction by European cultures. 

Just don't read books about europe.