Because of my husband's job, we must move all over JapanWe must move all over Japan because of my husband's job. We must move all over Japan because of my husband's job.

Sounds more natural to put the because statement at the end of the sentence.

Like my husband, wWe refer to those who must frequently be relocated because of their company's orders, such as my husband, and their families, as "tenkin-zoku" in Japan. We refer to those who must frequently be relocated because of their company's orders, such as my husband, and their families, as "tenkin-zoku" in Japan.

Like my husband, we refer to thoseI like to refer to people and families who mustove frequently be relocated because of their company's orders, and their families,for their jobs as "tenkin-zoku" in Japanese. Like my husband, I like to refer to people and families who move frequently for their jobs as "tenkin-zoku" in Japanese.

I was a little confused by the beginning of the sentence. By mentioning your husband directly, my understanding of your intent is that you personally like to refer to people with this term "tenkin-zoku". I edited your message to match this intent, since "we" is plural while "I" is singular. You seem to be talking about yourself and your preference separately from your husband's. If you remove the part at the beginning where you talk about your husband, then my understanding of your intent would have been that "we" (people from Japan in general) typically refer to people with this term. That this term is the Japanese word for people that move a lot for work. If this second reading is what you actually meant to say, then here is the correct form for that sentence: "We would typically refer to people and families who move frequently for their jobs as "tenkin-zoku" in Japanese." I don't speak Japanese so I didn't want to assume one intent over the other. Adding "like to" sounds more natural because it softens the intent and sounds more polite when talking about something you do. I would add "typically" instead if you are talking about "we" as in we people from Japan, since you personally can't speak for the preferences of all Japanese people. This also serves the same purpose, to soften the intent of your message. I also rearranged the words to flow more naturally in English. "to move" is much more common than "to be relocated." The second sounds stiff and clunky. Lastly I edited in Japan to in Japanese because my understanding of your intent is that this is the Japanese word for these types of people, referring to the language of Japanese not the landmass of Japan.

These days, in Japan, "tenkin-zoku" isare becoming unpopular because of the heavy burden. These days, in Japan, "tenkin-zoku" are becoming unpopular because of the heavy burden.

This was confusing in English because you have introduced this word as a noun (the people who move around a lot for their job). As an English speaker that only learned this term from this writing, that would make a literal interpretation of your original sentence to mean that people in general don't like people that move around frequently for their jobs. Was that your original intent? If that was your original intent, since you are referring to a group of people, you would use "are" (plural) instead of "is" (singular). Did you actually intend to say that the "action" of having to move around a lot for work is becoming unpopular rather than the people moving are unpopular? That people in general don't want to be a "tenkin-zoku" because of the heavy burden of having to move around a lot for their career? If you want to express this second intent: this is one way to write this: "These days, in Japan, having to be a "tenkin-zoku" is becoming unpopular because of the heavy burden."