Before the institution of Christianity and the evolution of Buddhism into Mahayana Buddhism, people never felt quite so strongly about religion. Suddenly, people feel ready to die for it and die trying to convert others, or even persecute others because of a different belief. What changed?
This question mainly relates to the time period in the question, but I have a feeling that the answer will lend some insights into the present situation in the Middle East.
Religion is a great cultural differentiator. People have been killing each other for many millenia, with a preference for targeting other people who belong to a distinct "culture", a rather loose term. From the outside, the god(s) people worship are quite easy to work out; if they are not the same as yours, then these people are "foreigners".
Historically, religion began to be a concept distinct from its host culture around 400 BC, at the time when true monotheism was invented. Before that period, there was no real difference, in the minds of people, between "your god is weaker than mine" and "your god does not exist"; early "monotheisms" such as Zoroastrianism and Mosaic Judaism were more properly defined as monolatrisms. The invention of the concept of monotheism came with the equally novel notion of your religion being something else than your ethnicity or culture; at that point, it became possible to die and kill for strictly religious reasons.
However, for persecutions to begin in earnest, it still required another ingredient: proselytism. In the Roman Empire, Jews were a troublesome component: conquest of Judea by Rome was not easy, and plagued by regular rebellions; but they were a people and their unrest was more political than religious. They did not try to expand their religion to other inhabitants of the Empire (especially after the destruction of the Temple in 70 AD: Judaism then restructured itself around the concept of orthodoxy, which de facto excluded the previously active non-mainstream branches which did try a bit of proselytism).
When Christianism appeared, it added the new component of proselytism, not only among Jews but also for all other humans (seemingly impulsed by the newly converted Paul of Tarse, against the advice of James the Just). Due to the exclusive nature of Christian worship, this lead to tensions with the Roman power, then persecution; and since the Christians were not defined by anything else than their religion (they were otherwise indistinguishable Roman citizens or slaves), the persecution was, necessarily, of a religious nature. Compounding the effect was the promise of an afterlife, with bonus granted in case of death when bearing witness of the new religion: that's what is called martyrdom. There is no martyr for Apollo or Jupiter, because when you die in the name of such deities, well, you still die, and you get no specific after-death benefit for it. With Jesus, martyrdom scores a VIP seat.
To sum up, you may get religious persecutions when all of the following hold:
The concept of religion distinct from culture and ethnicity has emerged.
A proselyte religion is active, leading to followers who are not distinguishable in a cultural way from the non-followers.
Dying in the name of the religion grants benefits which make it worth.
In the Mediterranean area, these elements were all present with the advent of Christianism. You also find them in the case of Buddhism at the time of the Great Anti-Buddhist Persecution (9th century AD in China).
In earlier times, people already felt strongly about religion; but no situation arose where the religion alone could possibly serve as justification for mass killing. For instance, during the Third Punic War, the alleged practice of human sacrifices by Carthaginian was a recurrent theme of Roman propaganda, and Romans found it abhorrent. However, since Carthage was a distinct polity from Rome, its destruction is not described as religious persecution, even though the cessation of the practice of human sacrifice was one of the goals of the endeavour. (Whether the practice was real in the first place is an orthogonal issue.)
From a conceptual point of view, people tend to define themselves relatively to some absolute notions which can be religion, ethnicity, social class... and will fight each other based on their respective stance with regards to these notions. Your feeling that people "did not fell strongly about religion" prior to Christian times may be an illusion due to the fact that religion had not previously achieved, by itself, the status of a war-fuelling absolute.
Jews had their quota of martyrs and persecutors too, it is even recorded in the bible. They have an exclusive and identity-defining religion too.
You are asking about persecutions, or to any situation where people are willing to kill or die due to religion?
Aztecs and their human sacrifices of enemies and the eternal wars required to supply victims; Huron x Iroquois have some similarity, although many prisoners were kept as slaves.
Phoenicians killing their own babies as human sacrifices;
Canibalism in general, in many cases is quite religious;
Pagans in most of Europe, Central Asia, (or Hindus in India even in recent times) killing slaves/wives at the funeral of their masters/husband to follow them in afterlife, or as a offering to the gods for the soul of the master; Oh, Vikings would gang rape the slave girl first. The last one strangled the girl still in bed.
The famous roman case of the oracle which requested "an offering of the best thing belonging to Roma". Then the leader got his best armor and weapons and jumped down a cliff because the best thing in Rome was "Men and Arms", and he himself was the most badass of them all.
Chinese emperor building great tombs and them killing all the workers so that the tomb remains hidden and his soul rest is not disturbed in the afterworld. (similar with Gengis Khan). Alas, when the Chinese Emperor can get away with killing whole groups of people, it is not partially because he has the "mandate of heaven"? Most European kings would be considered total tyrants and risk deposition...
Japanese commiting seppuku at command from the emperor, or just due to honor issues, which is partly religious.
Vestal Virgins which were killed if found to be not virgins anymore.
Aztecs also had a special offering of a young man from one of their ruling families. He had one year of 'paradise' with the best food and luxury and four ladies of his choosing (a coveted position). After that, he goes up the pyramid to have his heart taken... [A Cortes biography]
Some Brazilian native americans, had a kind of zodiac related to their gods, where people were assigned 'signs' related to the period when they were born. Some 'signs' were incompatible, and if a couple of incompatible signs would have sex or try to marry, both would be killed. [Gilberto Freire]
Caste violence in Hindu society is serious too. Killings and lesser attacks (like acid in the face of the low cast girl dating a high cast boy) still happens today. [see the news]
This without even start looking at muslim history, as you want BC stuff and not AD years, or at least civilizations not in close contact with christianity.
I wont also list many offenses not as serious as death, just some:
Greek/Roman oracles and taboos which changed their actions in strange ways, e.g. the Athenian generals which were punished (not sure if killed) after a great victory because they could not stop to fetch their dead floating in the water (big taboo letting corpses with no burial).
Some brazilian Tupi Indians, divided in tribes, they did not have much inter-tribe exchanges, but when they had a visitor from a related tribe, they would send their young girls to sleep with him as part of a religious welcome. Probably it was not so nice for the girl when the old fat Great-Uncle come to visit. [Freire or Sergio de Holanda]
So many people willing to be eunuchs in china.
Some of the examples above mix religion with political or societal issues, but is this not true also on any European religions conflict?
The point is that I do not think westerns are the only ones who take religion seriously enough.
And most of the examples above are serious violations of justice that were abolished with Christianity... Don't believe in Rousseau's "good savage"
But if you are asking why people are willing to die:
you would do anything at gun point?
Someone points a gun at your head, commands you, then you give money or information to a killer thug who will immediately harm other people; you help them to harm other people; you rape and kill your own family? No limits?
We may understand and forgive whoever folds and do whatever the thug wants, but the ones who did not fold were not the ones who did better, even heroically better?
If some people are willing to die before committing serious sin, aren't they right? If it is better (or heroic), not to betray your family even if you know they would forgive you anyway, then why it is ok to betray god?