Where was the cultural inheritance of the Greco/Roman world/antiquity preserved?

The "Greco-Roman" culture did not depend on Arabs to preserve it. Relatively few Roman works of science and literature are known only from Arab sources/translations. The literature of Greece and Rome was transmitted, as you said by the Byzantines, but also by many other sources including the Irish, the churches of Asia Minor and the Levant, the Romans themselves (to the extent they survived), the Christian Egyptians (the "Copts"), and the Christians in other parts of Europe, especially Gaul (France).

As just one example of this, one of the older and more important manuscript of Roman literature is the Codex Memmianus which has copies of numerous ancient works in Latin. This book is believed to have been written in Tours around 820 A.D. Another important manuscript is the Medicean Codex, which has all seven tragedies of Aeschylus, as well as books of Livy, Tacitus, Virgil and many other important works. This manuscript was written in the 10th century and appears to have been copied directly from original Latin and Greek scrolls of the 5th century or earlier. The earliest known manuscript of Pliny, one of the most important Roman scientific authors, is the Codex Moneus, which dates to the 5th century and was created and stored in a monastery in Austria.

The writings and literature of Ancient Greece and Rome were preserved in many places throughout the Middle Ages and beyond. The Byzantine Empire is considered to be the earliest heir to the Ancient Greco-Roman intellectual legacy. This of course is not surprising considering the fact the Byzantine Empire was essentially-(at least since the early 600's AD/CE), a Greek Christian Empire. For the Greco-Byzantines, the meticulous preservation of ancient Greek writings was an important part of their cultural and ancestral heritage. (Although the Byzantines identified themselves as, "Romans" and the earliest Byzantine Emperors were primarily of ethnic Greco-Roman or Roman ethnic descent, such as Constantine and Justinian, the Byzantines, as a society, a civilization and as a people, were primarily, a Medieval Greek Eastern Christian culture).

The Arabs played a very significant role in preserving ancient Greek texts and also helped to further the intellectual legacy of the Ancient Greeks, especially in the Sciences, Mathematics and Medicine. Works by Plato, Aristotle, Hippocrates, Euclid, Archimedes and many others were translated into Arabic during the Medieval period. Averroes/Ibn-Rushd, a Muslim Philosopher from Southern Spain, was one of the earliest Aristotelian thinkers and commentators in world history.

Roman architecture was both preserved and furthered by the Byzantines, as well as by the Arabs. The Byzantine Church Dome and arches were-(and are still) directly influenced by Ancient Rome. The influence of the Roman Arch can also be found in various Islamic monuments and buildings throughout Spain and Morocco.

The Italian Renaissance-(or The Northern Italian Renaissance), is considered to have been the Epicenter of reviving the cultures of Ancient Greece and Rome, especially in its birthplace, Florence-(though it also extends elsewhere in Italy, such as Venice). During the final years of the Byzantine Empire-(around the 1400's), small, but powerful groups of political, financial and academic elites-(primarily from Constantinople/(present-day Istanbul), escaped the onslaught of Seljuk Turkish conquests from the East. This small band of Byzantine Greek elites relocated to Venice and Trieste in Northeast Italy. They brought many things, including..... several classical Greek texts that were originally preserved in the University and Monastic Libraries in and around Constantinople. These ancient Greek texts from Constantinople were subsequently translated and distributed among the growing Italian intellectual and academic classes throughout Northern Italy. In other words, Byzantine Greek elites who fled the coming Fall of Constantinople helped to pioneer and shape the Northern Italian Renaissance, as well as the early Modern West. This is rarely discussed in most mainstream Early Modern History courses, though it is historically verifiable.

It would be interesting to make a list of principal ancient texts and how each of them reached us. And make a statistics. (Perhaps someone knows such a list?) Many of the texts that I know exist in both Arabic and Greek medieval versions.

Before this list is made, I want to express my doubts about Tyler Durden answer. He only gives examples of literary work (Tacitus, Virgil etc.) But another principal component of ancient heritage is science. The role of Muslim world in preservation (and spread!) of ancient science is very large. And the reason is that science was actually practiced in the Islamic world. (Unlike the West and the Byzantine empire). They not only preserved the ancient books but added lot of the results of their own research.

Apparently Islam was more tolerable to the ancient science, and people were not only allowed to practice it but were supported by the rulers.

Some scientific books were also copied in Byzantia, it is true. True but very strange. Just imagine this job: copying a book which you do not understand:-)