You could use a CAPTCHA.
Also, you can mitigate it instead by throttling their connection. It won't completely prevent them from screen scraping but it will probably prevent them from getting enough data to be useful.
First, for cookied users, throttle connections so you can see at most one page view per second, but once your one-second timer is up you experience no throttling whatsoever. No impact on normal users, lots of impact on screen scrapers (at least if you have a lot of pages they're targeting).
Next, require cookies to see the data-sensitive pages.
They'll be able to get in, but as long as you don't accept bogus cookies, they won't be able to screen scrape much with any real speed.