This is the same question that I asked myself about 2 months ago and I have done extensive research on the subject since then. (I lived between Livorno and Pisa for 3+ years so I am familiar with the area.)
First of all, how long are you in Livorno? One day or overnight? Could make a difference on how you approach the area. Time of the year can also matter.
Basically you have 3 options: (1) Ships excursion. (2) Private tour. (3) Self tour.
From what I have read, ships excursions seem to be geared around the one day in Livorno visit. Bus to the La Spezia area and water taxi/ferry to 4 of the 5 villages. Time in the villages is shortened because of the time it takes the boats to travel between the villages (as opposed to mere minutes by train.)
Private tour would possibly make sense if you are a party of 3 or more. The general mode seems to be drive you to the southern end and pick you up on the northern end. It varies by excursion provider.
The third is a self tour. You take the train both ways. This approach will work for those on a one day stop in Livorno but your time in Cinque Terre is somewhat limited. If you are overnight, and therefore have the luxury of a late night out, you have a lot more time. Train connections are quite good and train travel in Italy is quite easy (especially if you don't have luggage.)
I have two documents, in the advanced draft stage, on the subject. Let me know what you port schedule is and I'll post the appropriate one.
Radiance of the Seas - Nov 2006
Legend of the Seas - Jun 2007
Costa Fortuna - Mar 2008
Legend of the Seas - Oct 2008
Celebrity Constellation - July 2009
MSC Orchestra - Jan 2009
Navigator of the Seas - March 2009
MSC Poesia - November 2009
Jewel of the Seas - June 2010