My friend and I were having an interesting argument about the compilation of the Quran, especially on the topic of Muhammad's advice to memorize the verses as he told them to the faithful. We were wondering why exactly the order was given to memorize; we assumed that while the layperson listening to Muhammad wouldn't know how to read or write, as he was a merchant there must have been a scribe among them to record the matters, as he would any transaction.
Were there scribes among the Arab (or perhaps they aren't quite Arabs yet) traders? Was there anyone recording the Quran during its initial oral recitation? How exactly did we get the modern Quran from all those codices that were written and why exactly did Abu Bakr (or the other chap who's name currently escapes me) destroy them?
Early Islamic historians have conflicting opinions on the matter of the compilation of the Qu'ran. Various scholars would be willing to do away with the Islamic tradition you report; John Wansbrough, in a book approximately as controversial as it is unreadable, argued that the Qu'ran was actually elaborated over the course of more than a century, and that it was not the work of Muhammad. Wansbrough's position is quite extreme (and, in my opinion, quite conclusively disproven by findings such as the Sana'a palimpsest), but the existence of this kind of extreme position is indicative of the fact modern research has not reached a clear consensus on the issue of Qu'ranic origins.
As for your question itself, Islamic tradition itself mentions scribes, among which Ubay ibn Ka'b and Zayd ibn Thabit, in addition to a series of “individual” notes of anonymous provenance (al-Bukhari, for instance, reports a tradition about the sources of the Qu'ran: “parchments, scapula, leafstalks of date palms and from the memories [sometimes “breasts”] of men”). It might have been a misconception of the literate Abbasid world; but it is also worth noting that pre-Islamic Arabs were not living in an entirely illiterate backwater either. In fact, we even have a somewhat earlier example of high literacy amongst a group of nomadic Arab peoples living in modern Syria, the authors of the so-called Safaitic inscriptions (M.C.A. MacDonald, in various articles, even argues that they might have had one of the highest literacy rates of the ancient world).
I'm not sure I'd agree that there "must have been a scribe". The Arabian peninsula in the time of Muhammad was a predominantly oral culture; and, indeed, it remained so for an extended period following the Arab conquests, to the extent that it's quite possible to argue that not until the Abassids is there an easily observable "written Arab culture". So, Abu Bakr/Uthman/Umar didn't destroy codices; the Quran, as well as the vast majority of Muhammad's personal history as recorded in the Sīrat Rasūl Allāh, comes from compiled oral culture.