I'm a journeyman Electrician with 16 years in the trade......
1) GFCI receptacles are required on outlets within 6 feet of water, disregarding ANY appliances with a motor ( i.e.- refrigerators, disposers, diswashers, and laundry machines)...all kitchen countertop outlets are to be GFCI protected regardless of proximity to water.....these are referred to as "small appliance" circuits....
2) GFCI outlets are THE ONLY device that can be used to replace a device in an UNgrounded branch circuit, anywhere in the home.
3) Most if not all appliances with a motor, should be on their own SEPERATE branch circuit....(i.e.-refrigerators, disposers, washing machines, dishwashers, furnaces, etc....) the reason being that motors draw 125% of the required power needed to run upon start up.....think of your lights dimming for a split second when the washer changes cycles, that's the motor draw. ( say a motor runs on 16 amps, it needs 20 amps to start it)
While you may just be able to attach an appliance cord to the new dishwasher, and plug it in to an existing outlet, you MAY be overloading that particular branch circuit......depending on how old the home is, your kitchen may only have one circuit, maybe two at most......Figure that your refrigerator, microwave, coffee maker, toaster, etc....is already plugged into whatever circuitry you already have. In my experience, electrical fires are caused by one of three things.....1) Arcing- which is a condition where there is a bad connection, and the electricity actually has to jump a small gap in the wire, thereby creating a "hot spot" that can ignite....2) Overloading a circuit and 3) safety device failure.......Two and three kind of go hand in hand in that a breaker is a safety device meant to shut down or "trip" when an overload is sensed.....When the breaker is old and/or worn, it does not trip and open the circuit like it should, thereby allowing more current into the circuit than the wire can handle...This causes overheating, and in extreme cases, overheating to the point of ignition.
My best advice to you, if you are unsure, is to have a professional check out your wiring, and make a reccomendation. If you have an Electrician friend that you trust, ask him/her to do it. Any contractor that you call from the Yellow Pages is going to try and sell you some type of work. Keep that in mind, when you call. Also, remember, there IS NO HINDSIGHT after an electrical fire.......By the time you get to "I should of/ would of/ could of......" it most likely will be too late. Proceed with caution, and make smart decisions regarding your property and your family's safety.....Good luck!
PS- DO NOT trust anyone who is not experienced in this sort of work to actually DO THE WORK. Your homeowner's insurance policy will not pay out if you have an unlicensed goon burn your house down.