Well, you're already surrounding the directory you're searching through with double quotes, now try surrounding the $_ argument with double quotes too:
Thanks for the suggestion; I had that in my first version of the script -- but unfortunately double quoting $_ doesn't resolve the issue.
Rather, when the argument is supplied to grep it some how breaks the argument into to separate ones, in spite of the variable being double quoted. For example, if the $_ variable held Jon Doe -- the error would be:
grep Jon: No such file or directory
grep Doe: No such file or directory