Japanese has 熟語 jukugo, which is similar (if I understand the concept of a chengyu correctly): a set phrase of Chinese characters. The famous case are the 4 character set phrases, 四字熟語 yojijukugo, which are mentioned in the article, but there are other kinds too.
Besides that, there is 慣用句 kanyouku, which means just an idiom in general, and isn't restricted to kanji expressions. And 決まり文句 kimarimonku, which means something like a "cliche phrase".
Besides that, there is 慣用句 kanyouku, which means just an idiom in general, and isn't restricted to kanji expressions. And 決まり文句 kimarimonku, which means something like a "cliche phrase".