The Roman mythology we all learn about in school teaches us that Jupiter and Zeus are the same deity, etc. Stories that apply to Zeus are also taught as applying equally to Jupiter. That was the result of a deliberate project of Hellenization, and even in Hellenized Rome isn't exactly true.
But what about beforehand? What stories did the Romans tell about Jupiter and Juno and the other gods prior to Greek influence? Was there even such a time?
First, it's important to note that Roman religion was kind of a different beast than Greek religion and focused more on ritual than on narrative. So they were less concerned with developing a deep mythology behind their gods and focused more on pax deorum, their contract with the gods - offerings in return for favor. The co-opting of Greek gods and myths was in part due to the Hellenizing influence of Greece, but was also part of a pattern of co-opting other cultures' religions as a means of assimilating them into Rome, which was easier since Romans didn't really have a body of narrative mythology to go along with their gods (they could just say all your stories now apply to our version of your god). It's easy to point to the Greek gods and say they have the same story with different names, but that could equally apply to the gods and myths of other cultures Rome assimilated, met, or influenced (for example, Snorri Sturluson identifies Aeneas, more on him later, with a Norse legend). The Greek gods are just a lot more famous and have more stories written. Plus the influence of Greek culture in general is too noticable to ignore.
Additionally, it's difficult to answer this, as much of our knowledge of the founding of Rome is based on works written well after the fact, works that include so many aspects of Greek mythology that they leave the impression that Rome had always had Greek gods. For example, Livy, Virgil, and Cassius Dio trace the founding of Rome back to the end of the Trojan War and a man named Aeneas, who himself was a hero of Greek mythology, appearing in Homer's Iliad. Plutarch also mentions Aeneas as an ancestor of Romulus and Remus in his writings on Romulus, Rome's legendary founder and a son of Mars. According to Roman writers, Aeneas was a son of Venus, while according to Homer, Aeneas was a son of Aphrodite, supporting the conclusion that the link between the gods of Greece and Rome go back to before Romulus. In other words, Venus was Aphrodite before Rome was Rome. Based on this information, Greek influence would have been with Rome from the very start.
But Italy at the time of Rome's founding was dominated by the Etruscans, Samnites, and other civilizations, which means it's likely that early Romans, if they weren't following their own versions of Greek gods already, may have practiced some Italian religions with different sets of gods altogether from the ones we associate with Rome.
So what where the myths about Jupiter before he became just another name for Zeus? There are three options. His myths were lost to time. He and other early Roman gods were gods without any myths (a popular theory today supported by what we know about Roman religious, cultural, and political practices). Or he was always Zeus by another name (like historians wrote during Roman times).
This is answered in the FAQ (and has been answered many times)