After some poking around, I got Cyrillic to work in Calibre-generated MOBI.
Here are the steps. I have the original file in RTF, using MS Word (Windows PC), with all the desired formatting and ready to be "e-book-ized" (if it's a word). Then I do a "Save As..." and pick "html". The resulting HTM file then needs to be opened in an HTML editor - even the lowly Notepad will work. Toward the top of the source code you will see a parameter "charset=windows-1252". THAT'S THE PROBLEM. To display Cyrillic, it needs to say "charset=windows-1251". Change it, save the HTM file. Now import it into Calibre and do the HTM to MOBI conversion. (Look up "Character encoding" in Wikipedia for more scoop on "charset" values and what they do.)