There are many claims about the Mayan Long Count being more accurate than our own, but the claim seems to be simply copied from one source to another without anyone ever understanding it or explaining exactly how it's supposed to be more accurate. This claim seems to me to be part of the 2012 believers' efforts to build up the Maya into some sort of advanced race with great predictive powers as just another of their collection of false claims about 2012.
The Mayan Long Count is a count of days, unrelated to any natural cycle, so it's not possible to claim any degree of accuracy. Day 2 follows day 1. After 20 days a new period begins, so day 0.0.0.0.19 is followed by 0.0.0.1.0. Accurate? How would it not be?
The Mayan Haab or solar calendar had a year of 365 days, without any leap day added. Its error was therefore 1 day every 4 years. Our current calendar (Gregorian calendar) is accurate to 1 day in more than 3,000 years and is therefore far more accurate.
Some people claim that the Maya knew about precession, a movement of the of the Earth's axis roughly in a circle in a period of 25,800 years. Since the direction in which the Earth's axis is pointing affects the seasons, this has to be taken into account in an accurate seasonal calendar. Those who claim the Long Count was accurate say that the Mayans intended that a Long Count period of 7.13.0.0.0 or 1,101,600 days corresponds to two precession cycles with respect to the 365 day year. This does indeed yield an accurate value for the solar year, but seems to me to be playing with numbers rather in the fashion of numerology. Start with some numbers and the result you want to obtain, then perform various arithmetical operations on said numbers until they produce your result. There is no evidence that the Maya either knew about precession or incorporated it into their calendar.
