I typically oil the grill. After brushing it to get off any remnants from the last grilling session, I take a paper towel soaked in vegetable oil (something with a high smoke point), and rub my grill down with it. (I hold it using a pair of tongs).
I then let it sit for a couple of minutes, and then rub it down again it again -- you're effectively making a plastic coating, as you do when seasoning a cast iron pan.
Now for the unbelievable part -- the next time you run into problems with it sticking ... wait. I know it sounds crazy, but basically after a few seconds of grilling, the meat sticks to the grill ... after it's cooked long enough, it releases. Maybe not full releasing, but it gets ... less stuck. (assuming you're not using a sugary marinade that's turned into glue).
So long as you're okay with medium rare to medium, just give it another minute, and then try giving it a nudge to see if it's starting to release. You won't get the cross-hatch grill marks, but most people really don't care so long as it tastes good.