What a battle it was! But I won it at last. I finally saved my Russian file in every format known to humanity, and found that cyrillic type could be read in html format, or in Word97 .doc format, and in no other way. Using the Times New Roman font.
(deltame: my version of the EZ Reader doesn't come with the simsun.ttf font -- I found and downloaded it, and an rtf document shows the same garbage as Times New Roman. I'm too tired to try it on the html document, maybe it would work that way. Not bad-looking font, though -- the sans serif font my Reader came with isn't ugly, but it's not easy on the eyes for long-term reading.)
Thanks, everyone, for your suggestions!