Groberg served as a missionary of the LDS Church in Tonga. He experienced much difficulty in getting to Tonga: he was prevented from arriving by strikes, visa problems, and transport issues. Groberg served briefly in Los Angeles, Samoa, and Fiji while waiting for his transport to be finalized. When he finally arrived in Tonga, his first assignment was on the remote island of Niuatoputapu, which had had only limited contact with the outside world in the form of an occasional telegraph and a visiting boat. During the year he spent on the island, Groberg suffered from mosquitoes, a typhoon, and starvation. His missionary companion on Niuatoputapu was Feki Po'uha, who would later serve as district president in Niue, while Groberg was president of the church's Tongan Mission. After a year on Niuatoputapu, Groberg was assigned to more developed islands and served as a district president supervising smaller branch congregations of the church. Groberg later reported that the branches he dealt with lacked unity and morality. He had little contact with his supervising mission president and nearly drowned when pushed out of a boat during a major storm; he also suffered from exhaustion frequently. Groberg was denied an extension to his mission that would have allowed him to accompany a group of church converts to the New Zealand Temple. Groberg wrote a book about his mission from his memoirs called In the Eye of the Storm, which was adapted into the 2001 Disney film The Other Side of Heaven. The New York Times explains of Groberg's character, "The narrator and hero of The Other Side of Heaven, is a Mormon missionary dispatched to the Tongan islands in the Pacific Ocean immediately after his high school graduation in the 1950's." A sequel to the film, The Other Side of Heaven 2: Fire of Faith, was made in 2018 with the same actor, Christopher Gorham, in the role of Groberg.
LDS Church service
Groberg served for five years as a bishop in Idaho Falls. Groberg later returned to Tonga as the mission president and later as a regional representative. In April 1976, Groberg became an LDS Church general authority. In the mid-1990s, he was president of the church's Asia Area, where he was closely connected with the initial sending of church missionaries into Cambodia. He later served as president of the church's Utah South Area, where he was responsible for initiating programs for missionary work among the Latino population there, and attempts to ensure that English-speaking wardshome taught the Latino members within their boundaries, even if they attended separate Spanish-speaking congregations. Groberg also served as president of the NorthAmerica West Area from 1990 to 1994. In May 1992, Groberg presided over the organization of the San Francisco California East Stake, the church's first Tongan-speaking stake in the United States. In 2000, Groberg was called into the Sunday School presidency. In 2005, Groberg was designated as an emeritus general authority. From 2005 to 2008, he was president of the church's Idaho Falls Idaho Temple. Groberg's parents also served as president and matron of the temple from 1975-1980.