How to get Grill/Sear Marks
Grill marks are a sign of a really hot grate. As in really, really hot and well pre-heated.

You want your grate to sit over the hot coals for at least five, some sources say fifteen minutes. If your gril has a lid, close it!

Next, you need to ensure a proper heat transfer from the grate to your steak / burger / other food. You have two factors here:

a) Water will in fact insulate by forming a "protective" steam layer, so dry your meat. Use paper towels or whatever you prefer, just remove any surface liquid. This includes wet marinades, too.

b) Oil aids in the heat transfer, this is the reason we use some oil in a frying pan, too. You should always oil your food, not the grates, because you avoid flare-ups and the oil helps any seasoning - if used - stick to the meat. A light coating is sufficient.

Once you set the stage, you need to let physics and chemistry do their job:

Place your meat on the grill and wait. A proper grill mark needs some time to develop. Your meat will release itself from the grate once the "lines" are seared well. Do not rush this or your grill marks will remain on the grates as sticky residue, not on your steak. Optional: Turn your meat 30° or 45° to create a diamond- or cross-hatch pattern.
Flip, repeat on the other side.

Some practical hints:  


For those who love rare meat and thin steaks, there is simply not enough time to create a proper cross-hatch on both sides of the meat, possibly not even for a 45° turn. So rather aim to create one nicely defined set of marks than going for a mediocre pattern or a dry steak.
On the other hand, there are those who prefer a well-done brick of meat, which will not be at the desired doneness after searing both sides. Use indirect heat on the side of the grill to finish your meat until it reaches the proper temperature.
For everything inbetween: Note that every steak has two sides and one will be towards the plate. So serve your meat with the better looking side up ;-)