Is it safe to plug in three prong appliances into a three prong wall outlet that has no ground to it?

The NEC allows connection of ungrounded, 3-prong outlets if there is a properly installed GFCI "upstream" of the ungrounded outlet.

As long as everything is working as designed, nothing will happen, but that third connector is a safety feature to protect you. It is connected to the frame of the appliance and if there is an internal short, leads the electricity back to ground. Without that connection, the operator may become the ground, i.e., crispy critter.

No, it is not safe. Ground is a safety line, without it the appliance is not safe. It may work, and you probably will not be harmed, but it is not safe.