I thought this article was interesting, and it may be evidence that the early Persians had a method which might be seen as proving what they believed true (seeing as how these methods formed part of the instructional setting in which their form of mathematics is revealed). However, insofar as proof in its organized sense is missing, it might not qualify as disputing the claim that the early Greeks invented mathematical proof.
The real question that should be asked here is: what do we mean by the term: 'mathematics' -- or what counts as doing mathematics. Many of the responders seem to think that it requires the usage of abstract ideas, something, I'm assuming humans are capable of, and has to precede its invention.
In thinking about this, I would guess that the earliest mathematics probably involved counting, and possibly the assignment of numbers to a count (i.e., names given to mean a particular count, or possibly a reference to some count, or even an approximate count). I've seen evidence of counting in markings on rocks that seem to relate to the phases of the moon. Other evidence can be found merely by Googling this question. Seeing as how oral language precedes written, possibly by 1000s of years, there is probably no way to actually discern when such counting capability began.
Given this early evidence, though, is it the case that such counts, or even their having names associated with them, qualify it as sufficiently abstracted from their usage to make the claim that it is doing mathematics? (I'm assuming mathematics as a concept has to be distinguished from other activity.) I don't know. I even have a bit of trouble with the early Greek ideas in this area, some of which were confounded in the paradoxes of Zeno. Nonetheless, one would expect that within whatever ages we are speaking about there probably are a few smart individuals around that gravitate to making generalizations about what one is doing, enough to be called mathematicians anyway. Given this, I'd say mathematics might have been invented much longer ago that we have evidence of.
But, because there is undoubtedly a cultural or environmental aspect to this that precludes venturing too far, or, alternatively, provides sufficient freedom of thought to such folks, I'm inclined to think that the early Greeks are the ones who gave birth to the mathematics we know and love today. With respect to the how portion, I'd account for this, as Jay Bronowski does, in the particular environment in which the Greeks are situated where geometry becomes a key factor in measurement of the world it inhabits. much of it involving islands between which navigation becomes important.