This is an interesting question. The first thing to note is that there is a distinction between the Prophet Muhammed and previous prophets of God. Muhammed is special because he is Allah’s final messenger. It is true that there are many other figures who were prophets of Allah, but the Prophet Muhammed is not ‘just one’ of them: he is perhaps the most important of them all because, after him, there aren’t going to be any more.

Pre-Islamic Arabic religion was predominately polytheistic, with simple, needs-based gods to whom offerings would be given. The supremacy of Allah, the Creator God, was widely recognised, but so too were a multitude of other deities, including those drawn from Greco-Roman tradition. Although the importance of religious observance is hard to determine, it has been suggested that it is unlikely to have played a particularly major part in day-to-day life.^1 Life in the Arabian Peninsula was harsh, frequently violent, and often quite short: there were more pressing concerns than religion.

In addition to polytheism, there were significant external religious influences on the Arabian Peninsula. Christianity, Judaism, and Zoroastrianism were all present in Arabia to some extent, and there was evidently a high degree of familiarity with the Judeo-Christian religious tradition among the people of Mecca. Descriptions of the first revelation to the Prophet in the hadith — the large body of human-authored texts that accompany the divine revelation of the Qur’an — bear witness to this. According to the Prophet’s wife Aisha, after the first revelation in his dream, Muhammed was taken by his first wife Khadijah to her cousin Waraqah ibn Naufal: ‘a man who had become a Christian in the time of Ignorance’, and ‘wrote from the Gospel in Hebrew’. After hearing Mohammed’s story, Waraqah told him that ‘this is the angel Gabriel whom Allah sent to Moses’, explicitly identifying Muhammed as the latest in a line of prophets of a monotheistic Abrahamic God. Trading was a huge part of the economy of the Arabian Peninsula: Mohammed himself was a trader before he started to receive the message of the Qur’an, and it is likely that it was on trading caravans to places like Damascus and Jerusalem where he became familiar with the religious practises of Jews and Christians.

In short, Muhammed's Abrahamic prophetic lineage was something that was pretty clear right from the start, and it's unlikely that this aspect of Muhammed's teachings would have caused many problems: the Arabs would have been familiar with Judeo-Christian traditions, and the idea of a prophet of a monotheistic Abrahamic god would not have been new to them.

(1) For instance, Carole Hillenbrand, who writes that:

It is difficult to evaluate the importance of the religious practices and beliefs of the pre-Islamic Arabs. The Bedouin did not fight in the name of specific deities... such deities impinged very little on the everyday actions of the Bedouin, or on the rites and feuds of their tribal society. Life had simply to be endured with all its vicissitudes until, after his allotted span, man was struck down according to the inscrutable decrees of Fate.

Sources:

Carole Hillenbrand, “Muhammed and the Rise of Islam,” in The New Cambridge Mediaeval History, ed. Paul Fouracre. Cambridge: Cambridge University Press, 2005: 317–345.

This is a great overview of the context to the rise of Islam and is very readable. You may be able to find access to it online through a local library. I quoted from page 320.

Mahuna Muhammad Ali, A Manual of Hadith (1951). The bit that I referenced is on pages 7 to 8, and a pdf version is available to download online from www.aaiil.org