The group from the Tojolabal indigenous group in Chiapas were asked by armed village leaders to renounce their faith and it's claimed when they refused, their homes were burned - forcing them to flee.
They've been facing threats of this nature since April 2015.
Luis Herrera of the Coordination of Christian Organisations in Chiapas said municipal, state and federal officials were all made aware of the threats but failed to act.
CSW's Chief Executive Mervyn Thomas said, "We were deeply disturbed to hear of the forced displacement of these 34 men, women and children on account of their religious beliefs and even more disturbed that despite warnings for months, and a series of documented illegal actions on the part of the village authorities, the Mexican government, at every level, failed to intervene to uphold the law.
"The government should be ashamed that because of their inaction the situation escalated to violence and these families are now homeless. We call on the international community to raise this case and others like it with the Mexican government as a matter of urgency. We also call on Governor Velasco Coello and President Peña Nieto to take swift action to resolve this situation, to protect the basic rights of these families and to hold those who have committed criminal acts in the name of religious intolerance to account in the courts."