The frescoes were looted from the Mediterranean island more than three decades ago.
Archbishop Chrysostomos II said the Menil Collection plans to return the Byzantine frescoes early next year after the church insisted that they not "allow them to remain there even for one second longer."
"I salute this decision by the Menil Collection because embarking on a court battle would honor neither us nor the Collection," the Archbishop said.
Antiquities smugglers looted the frescoes from the Ayios Themomianos church in northern Cyprus following a 1974 Turkish invasion that split the island into a Turkish-speaking north and a Greek-speaking south.
Menil Collection founder Dominique de Menil obtained the frescoes in 1983, and struck an agreement with the Cyprus church to keep and exhibit them at a purpose-built chapel in Houston.
For more information, view the letter from the Menil director.