One term that inevitably rears its ugly head when talking about Mormonism is “Mormon Bible”. Usually someone who doesn’t know the first thing about Mormonism hears about how Mormons have their own scriptures and so assume that Mormons go around using a Mormon Bible. Now there is a half truth to this. Joseph Smith did go about creating his own “translation” of the Bible, a product of what he felt to be inspiration of the Holy Ghost since he wasn’t fluent in the original languages of the Bible and wasn’t working from ancient manuscripts to translate anything into English as translators usually do. This “translation” is known as the Joseph Smith Translation or JST and an example of this work, the recasting of Matthew, is found in the Pearl of Great Price and JST footnotes are found in Latter-day Saint church published Bibles.
With that caveat in mind, the translation of the Bible that English speaking Mormons use is the King James Version. Given that Mormons in non English speaking parts of the world using more modern translations, why are Mormons stuck with the KJV? Bearing in mind that the KJV is not the most accurate translation on the market (for example, the name Lucifer originated with a mistake when the Bible was translated into Latin and that mistake was carried into the KJV), and that it is difficult for modern ears to understand, especially when things get technical, some accuse the LDS church of choosing the KJV to keep Mormons purposefully in the dark. That is, a Mormon could potentially read over Romans in the KJV five times and not quite get it because of the language barrier. But if that were the case, why wouldn’t the LDS church choose a similarly difficult translation for use in non English speaking countries? No, that doesn’t explain it.
The explanation is simple: the Book of Mormon appeared in English (whether you believe it was translation or not) and the Book of Mormon language is very KJV-esque. More than that, the Isaiah portions of the BoM are largely a word for word match to the KJV. So you can see the reasoning here, unless the LDS church wants to redo the BoM into a more modern sounding English perhaps using a modern translation of the Bible as a guide, the KJV will probably need to stay around- people will expect those Isaiah passaages (among other things, such as the numerous paraphrased KJV verses found in the BoM) to match up with their Bibles. Since Joseph is believed to have been directly influenced by God in his translation of the BoM, messing with that (more than it already has been anyway) is unlikely to happen. This isn’t a problem for non English speakers for whom the BoM has to be translated into a different language no matter, and is modernized in the process to begin with.