I have a 3-gang box in my kitchen with one paddle switch and two outlets. Worked fine for 3+ years. All of a sudden the middle outlet is not working. No problems with the switch or the other outlet.

What might cause this problem, and will simply switching out the middle outlet solve the problem, perhaps?

It may be on another circuit from the receptacle next to it.
If a GFI is not tripped at the Panel, a Receptacle may be tripped in the kitchen.
Personally, I would have the switch at 1 end or the other, not between two cords/recepticles.

No GFIs are tripped at any recepticles or at the panel. The switch is on the end. The middle receptacle plug is the one that is not working. The plug and switch on the ends are both working fine.

An Electrician will remove devices from that box, test, and if not that box, they will test the box prior to that box.
Verify all GFI's in Kitchen will trip and reset.
Sounds like that is not a GFI and you have GFI"s leading me to believe it is protected by one. May be easy to determine as you are only supposed to have 2 recepticles per breaker max. which would be a GFI and a regular receptacle per circuit.