They are designed to have a reference to 0V ground. Many appliences with electronic components, particularly computers, can have quite serious problems when used on an old 127+127V supply in Europe so I doubt they'd be very usable on a US split phase supply either.
I'm running my PC and its monitor off of a 240V 15A (NEMA 6-15) outlet here in the USA, and there are no problems (the grounding (green) wire does go to ground). It's the usual USA split phase supply (120V-0V-120V). That doesn't mean that every computer and monitor ever made will work properly fed this way, but most all such use switching power supplies. Switching power supplies don't care if it's 60Hz, 50Hz or 400Hz or even 1.4 x RMS voltage DC for that matter. These supplies convert the incomming power to DC, and then to high frequency AC, then back to DC again (with isolation from the line).
Devices that use power transformers (like vacuum tube radios) will not have much problem with a slightly higher line voltage at 60Hz vs. the line voltage at 50Hz. At least the power transformer itself wont have any issues, the loads might get a bit unhappy but a well designed radio should tolerate 10% excessive line voltage from its nominial rating. We get the same problem with antique radios and tube hifi sets designed for 110V, and today's powerlines routinely provide 125V.
Devices like phonographs, clocks, and some motors will have problems with it being 60Hz instead of 50Hz. They will run 20% too fast.
