How can doc/docx files be converted to markdown or structured text?

Pandoc supports conversion from docx to markdown directly:

pandoc -f docx -t markdown foo.docx -o foo.markdown

Several markdown formats are supported:

-t gfm (GitHub-Flavored Markdown)  
-t markdown_mmd (MultiMarkdown)  
-t markdown (pandoc’s extended Markdown)  
-t markdown_strict (original unextended Markdown)  
-t markdown_phpextra (PHP Markdown Extra)  
-t commonmark (CommonMark Markdown)