This kind of thing happens to me all the time - but usually in the opposite direction, in that RTFs exported from InDesign hide or drop text particular kinds of styled text in other applications. Without opening up the RTF in a raw-text editor and staring at the tags (something I've been doing more and more as our translation memory tools mature, and the RTF gets comparatvely dustier and cobwebbier) it's hard to know exactly what is going on.
So, instead, I use a brute-force attack. If InDesign doesn't like Word's RTF output, I'd resave as .doc or .docx. If that didn't work, I'd open up in something like OpenOffice/LibreOffice/whatever the FOSS Office replacement of the moment is, and then just make a few minor changes and resave.
If that doesn't work, I would create a character style in Word and apply it to the text-in-italics to see if I couldn't change its format enough to get it past InDesign's Word/RTF filters.