A gas grill on a wooden deck is likely much less of a fire hazard than the stove in your house, so I would go with that. As to the gas versus charcoal debate, I stick with gas personally, simply because it is so much easier to use. Charcoal often has a different taste, but I didn't find that worth the extra hassle involved.

I also have a gas grill on a wood deck. No problems whatsoever.

We have a gas grill with a grill mat on our wooden deck.

They sell mats for those purpose. Check a hardware store.

Growing up we had a gas grill on a wooden deck. We now bring a charcoal grill into the backyard when we grill.

We want a grill, but we don't have a brick patio, just a huge wooden deck and a flat grassy area. We're thinking that we can't get a big grill for use on the wooden deck because of the fire hazard, and instead that we need to get a small grill that we can move onto the grass whenever it is being used. Does that sound right? Does anyone have any grill recommendations for our situation?

Also, gas or charcoal?