In answering this question, I'm going to assume that you're mainly talking about the Roman Empire during the Principate (from 27BCE to 284CE). The basic answer to your question is that Roman Emperors tended to not have natural born sons that lived into adulthood (or if they did, not long enough to become Emperor).

So let's take a look at the actual succession of Emperors. Augustus only had one biological child (Julia the Elder). However, Julia had 4 biological sons (one who died in infancy and another who was unfit for public life). The first two sons, Gaius and Lucius, were groomed by Augustus to become his successors. Unfortunately, both Gaius and Lucius died at young ages, forcing Augustus to designate his step-son Tiberius as his successor.

Tiberius himself had a natural son, Drusus, but he died at a relatively young age (supposedly murdered by Tiberius' Praetorian Prefect). Tiberius also had a few grandsons that were designated as heirs, but they were murdered in palace intrigues or by his successor. Skipping over a lot of the intrigue of the Julio-Claudian dynasty, Tiberius ended up adopting his grand-nephew Caligula and his grandson Gemellus (killed by Caligula shortly after Tiberius' death).

Caligula's own reign lasted only a few years and he did not sire a male heir before his own assassination, so his uncle Claudius was declared the Emperor by the Praetorian Guard. Claudius himself had a biological son (Britannicus) that he intended to designate his heir when he came of age, but Claudius' death/assassination and Nero's ascension prevented that. Nero himself didn't have any male children and after his death (and the year of 4 emperors), Vespasian came to power.

Vespasian, however, DID have two adult sons. Titus, the eldest, was designated his heir and ascended to the throne after Vespasian's death. Titus didn't have any male children, so his brother Domitian took the throne when he died. Domitian didn't have any children that survived to adulthood and after his assassination, an elderly, childless Nerva was declared Emperor. The next few emperors (Trajan, Hadrian, and Antoninus Pius) lacked male heirs, so they all chose their successor through adoption. Marcus Aurelius, however, did have a biological son (Commodus), and he raised him as his heir.

I'm going to stop there since this is already too long, but I think you get the general idea. Roman Emperors were rarely the natural born son of their predecessor simply because their predecessors lacked male heirs - which forced them to adopt someone else. However, when Emperors did have sons, they were very quick to ensure that their son would succeed them. In some cases this worked out fairly well (Vespasian and Titus) and in other cases it didn't work out well (Marcus Aurelius and Commodus). Anyway, hopefully this answers your question. If you're interested in more reading on the dynastic intrigue and general workings of the Principate, you can look at the primary sources of The Twelve Caesars by Suetonius and Roman History by Cassius Dio.

The short answer to your question is that blood ties were important throughout the history of the Roman Empire, but that often the children of the Emperor would not survive to succeed their father.

I am not going to cover the entire length of the Roman Empire, but instead focus on the Principate, as this time period involved less violent overthrow and more natural death for the emperor.

If an emperor is overthrown and killed then it is quite self explanatory as to why their children would not succeed them.

The first person to look to is Augustus himself. As he did not have any natural born children, he instead attempted to elevate his grandchildren (through his daughter) to the role of Emperor. Unfortunately for Augustus, both Gaius and Lucius died before their grandfather and Augustus chose to elevate his stepson Tiberius, as he no longer had any male natural heirs. There is speculation that this was a part of Livia's design, but again it's just speculation.

Tiberius, unlike his stepfather, did have a natural son to succeed him. Drusus, being the obvious heir would have succeeded his father if he had not died. Caligula, who was a blood relative of Augustus through Octavia, did not have any natural sons to succeed him and was replaced by Claudius.

Claudius did have a natural son Britannicus, who died before Claudius. Britannicus would have ascended to as co-emperor with Nero, but died prior to his elevation. Nero was then himself killed or killed himself and was replaced by a series of emperor, known as the year of the Four Emperor's.

In the end Vespasian was able to take control of the Empire. Finally you have an example of where the emperor dies with a natural born son who is still alive. The result is that Titus does become Emperor.

The next time that there is an emperor who dies with a natural son to succeed him is Marcus Aurelius and his son Commodus. The death of Commodus would lead to the year of the five emperors, which brought Septimius Severus to power. Upon his death, both of his sons, Caracalla and Geta were elevated.

The intention of going over those emperors is to show that whenever there was an emperor who died with a living natural son, that son would always succeed their father.

This trend would, with few exemptions, throughout the history of the Roman Empire. The ancient world was a dangerous place for a young heir to be.

Now as to whether family line was as important to the Romans as it was during the medieval times, yes. It absolutely was. You have to keep in mind that in the Roman Empire, the title of Emperor was not an office or a position that existed on its own. In fact for much of the empire, the Emperors were not even called emperors, holding a honorary title more equalling "Princeps Civitas" (First Citizen), only later taking the more powerful "Dominus" (Lord).

When an individual was elevated to what we think of as emperor, the position that they were given was head of household (or patriarch) of Caesar. The power, land and clientele was passed from one patriarch to the next. So even though Augustus and Aurelius were not by blood related, by tradition they were both patriarchs of the house of Caesar.

I hope that this has been helpful and please let me know if you have any questions.