If you like biking and love your bicycle, I would suggest bringing your own machine.
I am not fond of the combination of big cities + bicycles, so if it were my trip, I would use the three cities as bases for going to nearby bike-friendly attractions. For example, in Amsterdam, I might sightsee for a couple of days on foot or by tram, leaving my bicycle in the hotel room or storage, and then head out by train with my bike to someplace special, like Texel Island or Utrecht.