Were there any real cultures that actually worshipped the Titans, to the exclusion of the Olympians? Or perhaps mythical cultures that did so?
According to Hesiod, by the time the Olympians came to power, Golden man had vanished. Hugh G. Evelyn-White's translation of Works and Days (from ellopos.net) describes them as
(ll. 109-120) First of all the deathless gods who dwell on Olympus made a golden race of mortal men who lived in the time of Cronos when he was reigning in heaven. And they lived like gods without sorrow of heart, remote and free from toil and grief: miserable age rested not on them; but with legs and arms never failing they made merry with feasting beyond the reach of all evils. When they died, it was as though they were overcome with sleep, and they had all good things; for the fruitful earth unforced bare them fruit abundantly and without stint. They dwelt in ease and peace upon their lands with many good things, rich in flocks and loved by the blessed gods.
They were loved by the gods, but nothing is mentioned about man worshipping or serving the gods. It seems it was Zeus who first demanded that of the Silver man (same source):
(ll. 121-139) [...] But when they were full grown and were come to the full measure of their prime, they lived only a little time in sorrow because of their foolishness, for they could not keep from sinning and from wronging one another, nor would they serve the immortals, nor sacrifice on the holy altars of the blessed ones as it is right for men to do wherever they dwell. Then Zeus the son of Cronos was angry and put them away, because they would not give honour to the blessed gods who live on Olympus.
And then on it goes to the Bronze and finally the Iron age.
At least according to Hesiod it seems the Titans didn't need worhip, at least from the Golden man.
As for real people worshipping Titans, I can't think of a culture that worshipped Titans exclusively, but Cronos had a temple at the foot of the Acropolis in Athens, so at least one Titan was worshipped by real people, as cosmovision states:
Cronos ne figure guère que dans les mythes théogoniques; pourtant il avait un temple à Athènes au pied de l'Acropole.
Cronos hardly played a part in mythology except in theogenic myths; and yet, he had a temple in Athens at the foot of the Acropolis.
And of course there is the temple of Saturn in Rome (Wikipedia) and the Roman Saturnalia (Wikipedia), as well as the day after Friday. Saturn is usually seen as the Roman association of Cronos, so at least that one Titan did get worshipped quite a lot by real people, although, admittedly, it seems never to exclude the Olympians.
There used to be a theory that the Titans were actually the gods of the inhabitants of (geographical) Greece before the Greeks invaded and took it over. The idea there is that the story of Titanomachy is actually an allegory for the Greek takeover of their modern homeland.
This theory was popular enough back when I was in school 3 decades ago that it was presented to me as fact by an anthropology professor. The best online reference to it I could find today is from an online copy of The Minoan-Mycenaean Religion and Its Survival in Greek Religion by Martin Persson Nilsson (1950).
According to Mr. Nilsson, the identity of those chthonic Titan-worshippers would be the Mycenaeans. There's no reason it couldn't be older than that though. Knowing the history of the area, I'd think the Anatolians (eg Hittites) would be another possibility. Backing this up a bit is the fact that some Titans are loosely associated with Asia Minor. Another good possibility is the original non-Indo-European inhabitants of the area (who we have little record of, aside from the Minoans and some other nearby islands).
I'm guessing this theory is not currently in favor though, based on its vintage and the trouble I had looking it up online.
If we only consider humans cultures, there is not really any human culture before the gods of Olympus came to power. Prometheus is the one that created humankind and then gave them fire (which represents the superiority of humans over mere animals), as written here:
PROMETHEUS was the Titan god of forethought and crafty counsel who was entrusted with the task of moulding mankind out of clay. His attempts to better the lives of his creation brought him into direct conflict with Zeus. Firstly he tricked the gods out of the best portion of the sacrificial feast, acquiring the meat for the feasting of man. Then, when Zeus withheld fire, he stole it from heaven and delivered it to mortal kind hidden inside a fennel-stalk.
So since the humans only knew the Olympian rule, they never had to fear the Titans and worship them. They were thought to be powerless after their defeat in the Titanomachy
The war lasted ten years, but eventually Zeus and the other Olympians won, the Titans were imprisoned in Tartarus, and the Hecatonchires were made their guards. Atlas was given the special punishment of holding up the sky.