I was told that at the time around 1 B.C. to 2 A.D Greeks were still cremating bodies of their dead relatives. When did the Greeks start burying them?
This is a common misconception, cremation was not universal in ancient Greece. The Greeks had various funerary customs, that depended not only on local practices and customs but also on the social status of the deceased. Cremation was fairly common, however that doesn't mean that burial wasn't. In fact the more common practice in post Mycenaean times was cremation and burial, the remains would be cremated prior to burial. Inhumation was also practiced with varying degrees of popularity, depending on the era and place.
Cremation was known to the Greeks since at least the Homeric era. The Iliad has several mentions to the practice, every burial mentioned involves cremation with the more notable being the elaborate funeral games for Patroclus. A number of urns and jars containing burned bones and ashes have been discovered by Schliemann and others in Hisarlik, and seems to corroborate Homer's descriptions of the era's burial customs. However limited evidence of inhumation were also found by later excavations that show cremation, although popular, was not exclusive.
The popularity of the Homeric epics lead to the association of cremation with the era of heroes, however the Mycenaeans didn't seem to favour the practice. Most remains found in tombs of the era were not cremated, and the limited traces of charcoal that have been found in Mycenaean tombs have been attributed to funeral rites or fumigation, rather than cremation. The only conclusive evidence of cremation during the era was a jar containing burned remains that was found in a tomb close to the Heraion at Olympia.
Evidence of cremation are far more common in post-Mycenaean times, however it wasn't until the Archaic period that the practice became popular. It was practiced alongside inhumation, its popularity varied wildly from place to place and was almost always followed by burial of the remains, in group or single graves. The Phoinike necropolis gives us a unique opportunity to observe burial customs from the middle Classical period to the late Hellenistic era, and provides us with evidence for both practices, throughout Ancient Greek History.
During the late Classical period cremation's popularity generally declined in comparison to the Archaic period. One theory for its declination is that it was considerably costlier. There is certainly a general tendency towards simpler burial customs during the Classical period, pottery grave goods became far more common than metallic ones, however the cost of cremation and the cost of inhumation at the time is unknown.
During the Hellenistic period, cremation's popularity appears to have declined since the Archaic and early Classical period. As an example, of the 70 graves in the Hellenistic cemeteries in Thesprotia (Gitana, Elea, Doliani, Dimokastro) 42 were inhumations, 22 were cremation burials, 4 contained both cremated and inhumated remains and we aren't certain about the remaining 2. Similar patterns appear in other Hellenistic cemeteries, with some containing no cremated remains at all. On the other hand, most of the remains in the royal tombs in Vergina were cremated, and that is perhaps a hint that cremation was reserved for royalty and higher aristocracy during the early Hellenistic period.
In Roman Greece, the popularity of cremation further declined. It was ultimately replaced with inhumation during the 2nd century A.D., both in Rome and in Greece, mostly because of the advent of Christianity and by the 5th century A.D. cremation was abandoned throughout Europe.