GFCI
Many older appliances are not GFCI-compatible. A GFCI is set to trip when the current leakage to ground is in the range of 4-6 mA. Many older appliances exceed this amount. It is not necessary by today's code to have GFCI protection on a built-in DW, but countertop outlets are required to have it. If the DW is a built-in, I would recommend using it without a GFCI. If it is a portable, I'd say install a non-GFCI outlet under the sink, and a cord long enough to reach. That's cheating, but what else are you gonna do?
Whether you try to operate the DW with or without the GFCI (especially without), make sure to install a grounding cord. Dishwashers were among the first appliances that were required to be grounded, not only because of the water but also because of the close proximity of grounded plumbing. Make sure to verify the ground with a continuity tester or ohmmeter.
Enjoy your appliances safely!