I have been using a wire brush all my life with no stray "hairs" on any food. I don't use the cheap grill brushes sold specifically for that purpose, but a wire brush found in a hardware store. They are stronger, have more bristles and never shed. They are also stiffer and more dense therefore doing a quicker and better job. If you are still worried about bristles on the food, simply wipe the grate with a paper towel after wire brushing it to insure there is nothing left behind.
The other thing I do is after cooking the food, I leave the grill on high for a couple of minutes which burns off all the sauces, excess food, and dirt - makes cleaning with a brush a snap.