A military official said the bodies of 24 men and seven women had been found in the grave.
An anonymous tipster phoned in the location.
Officials suspect the number of bodies may rise as the recovery effort continues.
There was no information regarding the cause of deaths. However, the area where the grave is located has seen violent fighting between the Gulf Cartel and the Zetas gang as the rival drug gangs fight for control of Veracruz.
2014: The Year in Music [PHOTOS]