If any readers from any of these countries would like to post additional information or personal stories of conversion, that would be fabulous. Just add them to the comments, and I will link them into the article.
Patona Roa, the elder priest of the Maori people in New Zealand, fasted, prayed, and meditated for three days about which of the Christian churches his people should join. After his fast, he reported to his people that the true church had not yet come to the island, but that they would recognize it when it came because the missionaries would travel in pairs, would come from the rising sun, would visit people in their homes, would learn their language and teach in their own tongue, and would raise their right arms when they officiated. Within a few years, the missionaries did come, and within eight years there were 70 branches of the Church in New Zealand. By 1892, 10% of all Maoris were LDS. Presently (February 2017) there are 530,000 members of the Church in the islands of the Pacific Ocean.
Missionaries have proselyted in the India subcontinent as early as the 1860s, but only recently has the church had a permanent presence in India. The India Bangalore Mission was created in January of 1993 (October 1993 Liahona). Now there are nearly 12,700 members in India (MormonNewsRoom).
Although the church was established in South Africa in 1853, the gospel was not preached to black Africans until after the revelation on the priesthood (Gospel Pioneers in Africa, August 1990 Ensign).
(If you or any class members are troubled by the delay in extending priesthood blessings to Africans, please see my blog lesson Continuing Revelation.)
Post-revolution Eastern Europe – 1990 (Russia)
The Berlin Wall came down on November 9th, 1989 (Apr.’91 Ensign, p.26). The Saints in Czechoslovakia had spread the Gospel undercover during the 40 years the wall was up. In 1989, when the missionaries were able to return, they didn’t have to tract because there was a huge group just waiting to be baptized. As for the Saints in East Germany, within 27 years, everything President Monson (then Elder Monson) had promised had been received, including, in 1985, the dedication of the Freiberg Germany Temple – a temple behind the Berlin Wall! The Freiberg temple was so busy that patrons had to make appointments to participate in an endowment session (Ibid., p.52).
"In September 1989, Church leaders authorized a United States Embassy worker in Russia to begin holding group meetings in his apartment. Four months later, in January 1990, missionaries arrived in Leningrad. The first convert they baptized also became the first full-time missionary from Russia, who served in the Utah Ogden Mission. In February 1990, a congregation was organized in Vyborg. By mid-summer 1990, the Leningrad congregation, created in December 1989, had 100 members, and the Vyborg congregation had 25 members. In September, the St. Petersburg congregation was recognized by the government and in October a religious freedom law was passed. With membership in Russia at 750 in February 1992, two other Russian missions were organized.
"In June 1991, the Mormon Tabernacle Choir received publicity 'beyond its wildest expectations' as it performed in the Bolshoi Theater in Moscow and in Leningrad (now St. Petersburg). The choir recorded songs later broadcast to a potential audience of 339 million. In May 1991, the Church was officially recognized by Russia." Currently there are 100 congregations in Russia, with 22,720 members (included in the European number noted above) (MormonNewsGroup).
In the Middle East there are 4 congregations in Israel (I presume they are filled with BYU study-abroad students as proselyting is not allowed there--correct me if I am wrong), 5 in Turkey and a stake center in United Arab Emirates (thanks to reader Carl H for sharing this link! ) which was dedicated in February of 2013.
There are 11 congregations and a mission in Armenia. In great measure, Armenia was opened to the church because of the great efforts of LDS member Jon M. Huntsman, Sr., who built a concrete factory there which was staffed with LDS service missionaries and local Armenians to build safe homes after the devastating earthquake of 1988 which killed 50,000 people and left half a million homeless. The first member of the church was baptized in 1992, and the church was registered in Armenia in 1995 (MormonNewsRoom).
LDS military members also create a church presence in the Middle East. The Afghan Military District of the church was created in 2008. I don't have current statistics, but in 2009, there were more than 400 members in 4 branches serving across Afghanistan.