I usually do the same thing as the fellow with the burnbox, except I jsut do it in my cheap Wal-Mart grill. Get a bunch of wood sticks about the same size (one or two inches), either split them down from firewood or rip them on a bandsaw. Make a big old fire in the grill, and let it rip for a half hour or so until it burns to coals (enjoying a beverage while it happens). Once the fire's right, all coals, spread them out and put the meat on. I even managed to grilll up a couple tri-tips on Christmas day between rain squalls this year. I usually cook over madrone, apple or manzanita coals.
While I'm burning wood, wife puts a paste of salt, pepper, and some sort of other stuff on the meat. Agreed on getting the meat to room temperature before grilling. Sear the meat over a hot fire, all sides of it, then raise it up and let it cook under a closed lid until it's about 130 degrees. I try to let the slow cook part happen with the fatty side up. Let it sit about ten minutes, good stuff!
This year we had too much for the family to eat, so we had to take a bunch of leftovers, slice them really super thin when cold, then cooked them up a couple days later with some au-jus and some grilled inions for tri-tip dip sandwiches. I beleive my wife liked it better that way than eating it sliced hot off the grill.
When I take the meat off the grill, I close the lid and shut down all the vents. I invariably end up with quite a bit of leftover charcoal, whcih I scoop into a bucket for sometime when I don't want to mess with wood, or to add when the cooking takes longer than I planned.
and wishing they could stay, 
That would make some mighty good charcoal!
