"Bringing a North American extension cord and using it in Europe means you'll be running 230 V through an extension cord designed for 120 V. I'm no electrician but I'd feel a lot better using a cord bought in Europe."
The main issue with using a North American extension cord in Europe is the insulation ratings as that's what will be affected by the higher voltage. All extension cords sold should have at least U.L. certification, and they're most likely designed with a "safety factor" which means they can withstand higher voltages. Some of the European extension cords I've seen appear to have about the same thickness of insulation as the North American versions. A "flashover" in an insulated conductor is highly unlikely at either 115 or 220 volts.
However, given some of the products that are being imported from the far east, it's possible that a few "corners have been cut" in the materials. I've seen some very cheap electrical components in local hardware stores that were manufactured off-shore, and was surprised that they passed certification.
If the extension cord can operate safely with the higher voltage, it's not likely that the current will be a problem since it will be considerably lower at 220 volts to provide the same wattage.
One final point to mention. I only leave appliances (or extension cords) connected while I'm in the room and don't leave them plugged in while I'm out touring.