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.
In 1901, missionaries went to Japan, where they experienced a very slow start to the work, but successfully translated the Book of Mormon (Encyclopedia of Mormonism). By 1951, there were 30 regularly-attending Church members in Hong Kong. By 1960, there were 1700 members. Now (2017) there are 24,000 (Mormon Newsroom, accessed March 2017).
Temples were dedicated in 1980 in Japan, in 1984 in Taiwan, in 1985 in South Korea, and in 1996 in Hong Kong, and in 2000 a second temple was dedicated in Japan.
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 2 missions in India (MormonNewsRoom, accessed March 2017).
Presently in the 19 countries designated as "Asia" by the Church, there are 42 missions, 8 temples, and 1.1 million members (Mormon Newsroom, accessed March 2017).
(Previous information I posted on Asia from "An Ensign to the Nations" video appears to have been changed with further research.)
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 with about 30 active members besides BYU students and embassy workers (see reader comment below), 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.