A couple of weeks ago I outlined what sites and tools you can use to find frequent flyer award seats.
You almost always want to do the research to find award space yourself and then set up the award (online, if your mileage program offers all partners on its website, or over the phone).
One key tool I mentioned was Expert Flyer which I said had the most limited coverage of airlines but is indispensable for American flyers (because it has upgrade space) and because you can set it up to keep searching for space and email you when it opens up.