I'm not widely read in history (though I've been improving of late), but for what it's worth, my favorite book on history "beyond Greece and Rome" so far has been Charles C. Mann's 1491. In fact, it's one of the best and most illuminating books I've read, period.

It has about the information/idea density of a Neal Stephenson novel (a good one), and is about equally a pageturner.

I've also adored all of Jimmy Maher's (better known as The Digital Antiquarian) amateur history books, his Wonders of the World series - I think the one on Chinese history was especially ambitious and nice.

I enjoyed Toby Wilkinson's The Rise and Fall of Ancient Egypt, but it's a bit one-note; still looking for the Egyptian history book I'll fall in love with, though as as a general map it's quite good.