If you need this only once, and you don't mind it being slightly wrong, you can use a regular expression to hunt for comment start indicators, and simply count the number of hits. My unix is bit rusty, but I think you want something like
grep -r '.*/[\*/]' yourdirectory | wc
This finds lines that apparantly have a comment start in them, and then "counts". wc produces three numbers: line count, word count, character count; the first value is line count and corresponds to the number of hits, therefore the number of comments encountered.
If you want the number of comment lines, this will badly undercount multiline /* ... */ comments; you'll get a lower bound. But then, you wanted a fast answer.