I think it might be a bit of a misunderstanding to suggest that there were no clear laws of succession in the Roman Empire; there was. In the Principate era, this was based on the same laws of succession that occurred with any familial based succession. Your eldest son would often inhereit the role of paterfamilias, but any lands and property you controlled would have to be set out in a will. The Roman Emperor in this time was not an official position, and his power was based on his vast wealth and the direction of that wealth into control over the legions. Through your will and the familial connection your heir would inherit that and thus take his role as emperor de facto, while his role as emperor de jure would be bestowed upon him by the Senate.

However, in practice, the 'role' of Emperor, even down to the Byzantines, was never really formally established in very concrete terms, and the succession law was thus not focused on. Instead, the succession was determined by strength of arms. Those that could control more troops from their challengers were the ones who could remain on the throne, even to the Byzantine era. The only formal attempt to quash this was Diocletian's Tetrarchy, an experiment in formalization of the office and succession that failed withing a decade of it first being tested.

Spoonfeedme's explanation is correct in that acclamation of the Roman legions (and immense wealth) was often the surest way to claim the principate, I don't think that quite answers the question as it is not really a true method of peaceful succession.

My current graduate Latin instructor always describes the Roman Emperor as possessing a bundle of powers. By this this I mean that the emperor held most of the traditional powers of the offices of the old cursus honorum. Therefore, since the powers of the emperor had their roots in constitutional offices, they could not be inherited. This aspect of the emperor's power is implicit in the res gestae divi augusti (http://classics.mit.edu/Augustus/deeds.html). Although the emperor's power was not limited to roles of these traditional offices as the principate progressed, but traditional elements of the Roman constitution persisted even as the emperors gobbled up powers.

That doesn't mean, however, that there were attempts at instituting some sort of succession. I once saw Dr. Edward Champlin of Princeton University give one of the most interesting talks I have seen in which he attempted to recast the role of Sejanus. The goal of his argument was to see past the invective of the Roman historians that uniformly vilify the Roman figure so ably played by Patrick Stewart. Champlin suggested that the assumption of magistracies was not simply the act of a power hungry politician, but rather the result of Tiberius grooming him to be his successor. There was no roadmap for doing this, but towards the end of his life, Augustus began granting more and more powers and offices to Tiberius, and there's good reason to believe that the same thing was happening to Sejanus. Champlin also pointed out that Sejanus began to cultivate relationships with the old Etruscan families of power and the major poetic circles in the same way that Augustus had done.

Spoonfeedme's example from the reign of Diocletion is a less subtle example of an attempt to establish succession laws, but it still failed.

Unfortunately, I must plead innocence on succession formalization under the eastern Roman Empire. my detailed knowledge of that aspect of Roman history is limited to the series of concessions made to Latin powers following the restoration of the Palaeologoi to the throne after the Treaty of Nymphaeum (1261).