Jehovah's Witnesses calculate Nisan 1st as the new moon closest to the spring equinox and this actually does produce a workable luni-solar calendar. If you count the number of days from Nisan 1st to Nisan 1st by JW reckoning, you'll see a pattern of "Normal years" of either 354 or 355 days and "Long "years" of either 383 or 384 days. This pattern follows a 19 year cycle with long years occuring in years 3, 6, 9, 11, 14, 17 and 19, so it is actually a Metonic cycle in its own rite.
However it is also a strongly cyclical system. Memorials can occur as early as a day or two after the spring equinox or as late as the 19th of April. This also follows a 19 year pattern with "early" Memorials occuring in years 3, 11 and 19. In recent years, "early" Memorials occured in 2008, 2005, 1997, and 1989.
This cycle isn't a problem for Jehovah's Witnesses because Memorial is the only day out of the year they're really interested in calculating, but it would be problem for Jews because they have many other holidays throughout the year and once you've set a date for one holiday, you've set the date for all the others in that year.
In 2008 for example, Jehovah's Witnesses calculated Nisan 14th to be March 22, just two days after the equinox. Again, not a problem for them, but this would have pulled the 15th of Tishri back to the middle of September, which is a little too early for Sukkot. (--Feast of ingathering. In 2008, this began on October 14.)
The modern Jewish calendar moderates the cyclical effect which produces "early" Memorials with a wider range of year lengths. Normal years can be 353, 354 or 355 days in length and long years can be 383, 384 or 385 days. Holidays therefore occur in a correspondingly narrower range of dates. The difference between an "early" Memorial and Pesach is usually about a month.
I don't know exactly what system(s) the Jews in the Tannaitic period used and I'm not sure that anybody does. (I hope Leolaia will correct me if I'm wrong on that.) I do know that they would have run into the same difficulty I've described here affecting holidays directly tied to agricultural events if they had used the exact system Jehovah's Witnesses use to calculate Memorial.