Under compile, look at the Formatting tab. There you will see checkboxes for each of the file and folder levels. Uncheck the `Title' checkbox for your chapters. You should see in the preview field that the title goes away ... I have played with this on a sample file and achieved the results you want when compiling to a mobi file. (At first the title of the first document appeared, but that doc was not in a folder in the project - when i put it into a folder I got the results you desire.)
Note that all the documents in my test project have titles, some quite long. Just unchecking the Title checkbox kept the `Chapter One, two, three' et al but did not include the title in the table of contents, in the osX Kindle program.