If it's good, dry, hardwood (Oak, maple, ash, etc.) it should be fine. Even a little pine may be OK - gives it a bit of "campfire" flavor.
I would start by splitting the bigger logs down to the rough equivalent of 2" X 2" boards. Make sure they're the right length to fit on the charcoal grate of your grill.
However, I would usually burn-it-down to (at least, mostly) coals before cooking:
The best way to do this is usally a "log-cabin" fire, where you stack pairs of logs, alternating at 90-degrees. It looks sort of like the walls of a log cabin with an open space in the middle and four "walls". Use more wood than you think you'll need - it will burn down a lot. Once lit, this will burn-down quickly - producing a nice bed of wood coals.
Then, spread the coals and you're ready to cook!