How can I keep delicate food from sticking to the grill?

Other people have answered similarly but definitely make sure your cooking surface is VERY HOT before the food hits the surface. Cold food+cold metal=sticking!

In addition to this, you should also make sure you don't start trying to flip too early. Let the food cook and get a good sear on one side before you try to flip. If you do this with a hot surface, you shouldn't have any problems with sticking.

One more thing, it's important that the grill surface is clean, so cleaning your grill religiously after you finish using it each time, and then oiling it lightly before you use it, can be a huge help in this area.

I'll start with agreeing with the others -- oil your grill (after first cleaning it), and make sure it's hot. Think of it like seasoning a cast-iron pan, but you have to redo it, because the heat from the grill can bake it off.

Next, don't keep trying to move it. Things like chicken and steak will release from the (well oiled) grill once they've gotten a sear on them.

Fish, however, I go with an alternate, possibly controversial method -- aluminum foil. Use a piece of foil slightly larger than the piece of fish (one for each piece, you want space between each piece for the smoke to come up) and oil it slightly. When set onto a hot grill, you'll even still get grill marks.

I've found that if you brush on some cooking oil either on the chicken or directly on the grill it will help. Also, if you move the meat slightly before it cooks too long it will help it from sticking so much.

In addition to oil, make sure the grill is nice and hot before you start cooking. The same principle applies when pan frying.

Generally the best way to prevent sticking is with fat. One solution is to apply fat (oil) directly to the meat, or to the grill, right before cooking. The other thing is to avoid cleaning the grill too much. While you should keep too much junk from building up on the grill you do want to let it season, much as you would a cast iron pan. As the fats and everything build up on the grill it'll help future meals from sticking.

All the books I've read say that oiling your food is the preferred method.

Having said that, I often oil my grill by using a paper towel with oil on it

You could season the grill. Oil the grill before you start cooking and then heat up the grill. The heated oil will create a 'non-stick' surface on the grill. The more often you do it, the more non-stick the grill will become.

I'd been using paper towels to oil the grill, and I just discovered that newspapers work a lot better. Paper towels absorb the oil and then don't release it, and they leave fibers behind on the grill. Newspaper transports the oil but then releases it onto the grill.

I tried this yesterday and after several applications of oil, the grill was terrifically seasoned, to the extent that the food was sliding around a bit too much.

Something I tried for the first time today: I used oiled green onions to create separation between my fish and the wire basket. My fish came out perfect and the charred green onions improved the presentation and maybe even improved its flavor.

I am really quite surprised no one has mentioned grilling on wood planks yet. It's an extremely useful alternative to keeping the lighter meats/seafood from sticking to the grill surfaces. Typically they aren't too expensive and you get the added bonus of the smoky aroma in your food.

Link to a decent primer on wood plank grilling for those interested: http://backyardprovisions.com/blogs/grilling-101/10784049-wood-plank-grilling

Yep. As others have said, don't be afraid of fat or heat. Let it sit for a while. When the crust has formed and the food is ready to be flipped, it should release fairly easily with a bit of gentle nudging. Additionally, I find that dredging fish in a bit of flour helps to prevent sticking.