Here is a method we've used successfully, even in last year's torrential downpour.
Split into pieces about 2"x2" and start building 3 or 4 layers of a log cabin style fire (square, alternately stacked courses).
Split a couple of pieces into light, flat slabs and cover the top of the center of the fire area. This keeps the rain off the fire and center of the wood.
Split into fine kindling pieces for the fire itself. I often make a teepee style inside this log cabin. The trick is to work between the open areas of the stacked courses.
Light the middle, dry kindling (from center of wood, as others have mentioned). You may have to make shavings or shaved ends of wood- very thin like paper on one side, thicker as a 'handle' on the other.
As the fire burns, keep feeding it this very fine wood and kindling until the inside of the stacked courses begin to burn. When they do, use progressively larger pieces (until a full 1/4 split piece) to push the burning pieces into the center of the stack, taking its place.
As you get larger pieces built into the base courses, you can go up higher. The wood dries nicely in the log cabin style without hurting the fire. The pieces on the top can be exposed to full downpours and it won't hurt. The fire breathes out the space between the courses.
Don't allow the center red coals to burn down, you want to build them up. Poke the ends of pieces burnt through the middle deeper into the fire.
Once the fire is burning for some time and the red coals are built up to a good level, you'll notice the waavy heat inside the stack. Now it is ready for mountain pies!
Here's a trick we use for pie irons.
Carefully grab the two ends of a course on the side where you want to put the iron. Spread them out a bit and pull one of the cross pieces towards you. This will open up a triangle through which you can insert your iron and the cross piece and side piece make a nice corner to hole the handle/rod portion.
Make sure your stack can handle this weight or stack a few pieces outside the cabin upon which to lay the handle/rod.
During this time, we also stand our split wood on end, drier/split side facing the fire on the inside of the fire ring to dry out.
If there is a LOT of water and it puddles or runs into your fire ring, I start with a few pieces of pieces split into halves and laid flat as a platform upon which to build the fire and optionally stand the drying pieces upon to keep the ends out of the water.
We burnt a fire last Summer in the state park. The only fire in the campground. About a dozen neighbors' kids came back with DD and (after parents' permission) we made a ton of pizza mountain pies! It was a ball!
We had a couple of 'visits' from the young rangers...well, more like driving past to check for flooding and pausing, looking, waving, and probably shaking their heads, "Rednecks" :B
We went to bed late, it poured all night, yet there were enough coals we could've stoked the fire up in the am!
The fire was just a bit beyond the end of the awning. We sat under it, plenty dry, even holding the ends of the pie irons in relative comfort.
Hope this helps you enjoy your next rainy camping day!