How can I build a fence with posts that rest on concrete?
You could build a modified free standing Colonial Style Log Fence. It is basically logs or boards in a zig-zag pattern. Instead of logs it can be built with dimensional lumber 2x4's or 4x4's etc. They were originally used to mark property lines or animal confinement. You could drill a hole in both ends of the boards and use a piece of pipe or re-bar as a pin to align the zigs to the zags. The advantage of this type of fence is wind passes through it. It would take a decent gale to move it. The minus is it uses a lot of material per linear foot of fence line and depending on the lumber used could allow small animals or children through the fence.