Make it like an arbor or garden gate. You can landscape in front of it and put some climbing plants on the arbor. After this discussion that's what I think I'm going to do next summer. I'll do the rail fencing with the mesh inside to keep the dog in. At the end of the driveway I want an arbor for grapes and a gate. I'll have to put a gate large enough for a vehicle on the other side.
I'll have to look at my options...I'm going to check out those picket fence sections, which might be the easiest thing, and I think they have gates that set right into those. My biggest concern is that the snow plow tears up the first few feet of lawn every winter, so if I'm not careful (i.e. don't put the fence back far enough), I'll have to redo it in the spring...ugh!
That sounds really nice! It wouldn't look like a doggie run at all! If you put the doggie door in a wall instead of the side door, you'd still have access in and out the door without having to go directly into the run; just have a gate into it near where that door comes out. Where the yard begins to slope down, you could create a little deck area and put a dog house on it so that they would be out of the heat and dirt.