Part of the thing that complicates this is that the Mayan never really stopped using cities. They just used a different set of cities. The last hold out of the Mayans was the capital of the Itza Kingdom, Nojpetén (or Tayasal) that was only conquered by the Spanish in 1697. The place had been inhabited from the Middle Pre-Classical period (900-600 BC) though, it didn't really expand into a major city until the Early Classical period (200-400 AD). This means that this city survived the Classic Maya Collapse of the 9th Century just fine without being abandoned.
Most of the cities abandoned in the 50-100 years of disasters that make up the end of the Mayan Classical Period were in the Southern portion of Mayan territory. It seems that different cities were abandoned for different reasons: wars become more deadly and more common, famine and drought struck the region, trade routes shifted further north, and so on. It's hard to distinguish the migration myths from this even from those the rash of abandoned cities in the 15th century and earlier in the 1st century. What makes the 9th century collapse different is that people didn't resettle the newly abandoned city in a century or so as had been the case in both earlier and later collapses and building something entirely new was a great strain on an already diminished people.
In short:
It was a very long time ago and the people who immigrated later assimilated into other cities resulting in a mixing or loss of the traditions of the defunct city.
This wasn't the only collapse, it was simply the most dramatic one. Oral traditions didn't do a good job of distinguishing which collapse they were talking by the time the Spanish arrived.
Different cities were abandoned for different reasons. There wasn't one reason for the collapse, there were many. Many city sites were abandoned only for new ones to spring up very close by, but on more defensible ground with much stronger fortifications which would imply that more damaging wars was at least part of it, but this doesn't come close to explaining all the abandoned sites. The cyclical nature of Mayan collapse seems to indicate that they relied upon unsustainable agricultural practices, which resulted in famine and societal collapse every few centuries but that wouldn't preclude resettlement after the soil recovered. No one factor seems to fit the bill as "the" reason, so it seems that the most reasonable conclusion that it was a mix of other reasons that came together in a certain way... which makes it hard to be definitive.
Outside of the Catholic Church, there was little to no effort put into preserving Maya Culture (and the Catholic Missionaries were only interested in so far as it supported evangelization) for several centuries after contact. A great deal of the culture was destroyed during and after the conquest.
For additional reading:
As an addendum to /u/A_Sporific's fantastic response, you may be interested in the section The Maya in our FAQ which contains several previous topics on the Maya Classic period collapse.
I want to also add that there is a disproportionate amount of scholarship on the Classic period. Both the public and academia are fascinated with the idea of lost jungle cities and mysterious collapses. Because of academic focus on the Classic period both the Preclassic and Postclassic periods have suffered from a lack of equivalent attention. This gives a skewed perception that the Classic period was somehow more populated, prosperous, and important than the preceding or succeeding periods. I've provided below several sources that you may want to consider reading if you would like to further explore the topic of the Postclassic Maya.
Chase, Diane Z., and Arlen F. Chase. "Framing the Maya collapse." After Collapse, The Regeneration of Complex Societies (2006): 168-187.
Masson, Marilyn A., Timothy S. Hare, and C. Peraza Lope. "Postclassic Maya society regenerated at Mayapán." After collapse: The regeneration of complex societies (2006): 188-207.
Jones, Grant D. The conquest of the last Maya kingdom. Stanford University Press, 1998.
I want to throw this out there for future reference, Mayan refers specifically to the language the Maya spoke. So it'd actually be Maya civilization, Maya people, etc. You wouldn't say Mexico speaks Mexican either.