Okay, I've got a bit more info on this. I hope you don't mind a little reverse-engineering...
The New Posts link has a few parameters on it:
search=site:disboards.com - This clearly directs the searching to just the DIS.
scope=0 - This search just for posts within the last 24 hours. Higher numbers search back further in time.
order=1 - This, I think, is a bit strange. A '1' here means "sort by relevancy". Switching this to '0' would sort by date. That might be a better choice under the circumstances.
do=getnew - This does the magic, I believe, choosing only threads that are new. I suspect BoardTracker added this just for the DIS, because I don't see how to do this from their own website.
I believe we can even limit the returned information to specific forums by adding some paramters to that effect on the URL. I've done so and it seems to work for me. So I'll keep my own URL for New Threads in my browser's bookmarks.
One other thing which is clear to me, but may not be clear to others, is that the results through BoardTracker are updated periodically. (Hourly maybe?) So if you hit New Posts again, it'll probably show you the same thing it did last time, and then every so often it'll show you a completely different set. I think this happens on the hour (at least) because I've noticed that the boards suddenly get a lot slower for a few minutes, right around 10:00 AM.