When Joseph Smith had his First Vision, or presented the Book of Mormon as his first public sermon, the message was clearly for the Christians of his day. Ever since then most missionary work has been geared toward those who recognize "The Gospel" as a set of particular theology. They might not agree on the specifics of the theology, but they would not misunderstand the message. As a group that would like to become a "World" rather than a "Regional" religion, Mormonism's success among non-Christian society has been far from promising. To the chagrin of many Christian denominations it seems that Mormons are destined to follow the coat tails of other sects missionaries. That might be so, even by God's design.
Still, it would be nice to think what could possibly be an approach that opens up more missionary work among Taoists, Buddhists, Hindus, and Muslims, etc. Is the message of the Restoration simply that? The Creedal Christians teach of Jesus Christ, and Mormons teach of a return to the original Christian authority. Looking at European Christians' recent response to Mormon missionary work and the "among the fastest growing" doesn't sound very possible to sustain. Ironically, it is in Africa that the growth has been most noticeable aside from the Western Hemisphere. Even then, it is mostly among those who have been taught Christianity and already have faith in Jesus Christ as Savior. Most of the rest of those who follow other religions are deaf to Mormon missionaries or don't have a chance to hear them teach.
The future of the LDS Church, as trends seem to indicate, is not among the Westernized Christians who are losing their faith and morals as the years progress. It is to be found among the non-Christians who are not afraid to declare God in public and base their lives off of their own prophets and Scripture. What can Mormon missionaries learn from other denomination Christian missionaries? Should we learn from them? It is beyond my personal observations and any kinds of expertise to even start answering those questions.
Ammon can be the start of a template in reconciling these difficulties. His first missionary activity was not to preach the word, but to live it in service:
22 And the king inquired of Ammon if it were his desire to dwell in the land among the Lamanites, or among his people.
23 And Ammon said unto him: Yea, I desire to dwell among this people for a time; yea, and perhaps until the day I die.
24 And it came to pass that king Lamoni was much pleased with Ammon, and caused that his bands should be loosed; and he would that Ammon should take one of his daughters to wife.
25 But Ammon said unto him: Nay, but I will be thy servant. Therefore Ammon became a aservant to king Lamoni. And it came to pass that he was set among other servants to watch the flocks of Lamoni, according to the custom of the Lamanites.
-- Alma 17:22-25
It is often said that Mormon missionaries are seen as simply uber-Americans for good and bad. There is plenty of truth in this. Now, according to LDS Scripture this is also what God intended. However, there is also warning that flagrant disregard for the reason God chose America as center of the Restoration would be disastrous. It might be prudent, God willing, that an Apostle actually live and work full time in an Asian or African country. Better yet, that a person from those places who are strong in the faith and yet come from a completely foreign background are called as Mormon missionary ambassadors. This would be for both directions - teaching the Gospel to one group, and customs, belief, and etc. to the Saints.
Ultimately, it comes down to what would interest a non-Christian to be interested in a blatantly Christian (regardless of sectarian creedal arguments) religion. Right now there are too many political, aside from religious or cultural, boundaries to make the question of immediate concern. Most of Asia and the Middle East are off limits to any Christian. Even the inroads that have been made between Muslims and Mormons are seriously compromised by equal amounts of zealous claims of theological truth. India is a country that, although still volatile, has the most promise for ripe missionary work.
The person that comes to mind most with these issues is the "Apostle to the Gentiles" Paul who seemed to arguably navigate between the Jewish world and the Roman one. His success, like what was pointed out as a suggestion above, seemed to be his dual citizenship in Israel and Rome. Better yet, his familiarity with both cultures. Perhaps what continues to make the idea of a greater non-Christian missionary effort so enticing is his proclamation:
14 I am debtor both to the Greeks, and to the Barbarians; both to the wise, and to the unwise.
15 So, as much as in me is, I am ready to preach the gospel to you that are at Rome also.
16 For I am not ashamed of the gospel of Christ: for it is the power of God unto salvation to every one that believeth; to the Jew first, and also to the Greek.
And it could be added; to the Asian, Euro-Indian, African, as well as the European. If Western civilization is slowly dying (as I believe it is from its own peculiar social isolationism) the question of missionary work among non-Christians is becoming more relevant. The LDS Church has a chance to greatly expand its numbers in an honest world-religion status. Such a situation can only happen when the Saints step back from its tiny larger shell of a shell and consider the greater picture. Having an expanded vision and different approaches (whatever they might be) could make "are Mormons Christians?" seem trivial and banal inter-sectarian fighting. Having success outside of "Westernized culture" could make for a greater diversity of faith and an expanded definition of Zion.