Trouble syncing multiple Apple products between multiple computers
If you really want to sync between multiple computers, it is technically possible. You have to place your iTunes library folder in a location that can be accessed by all of the systems on the network. Then, when you want to access that shared library, hold down the Option key (Shift key on Windows) while iTunes is opening. This will bring up a dialog box with an option to select another library. Click it, and browse to the shared network folder with the iTunes library in it. iTunes will open with this library, and you should be able to sync your device on that computer.

The downside? Unless you have a server or network attached storage device, the computer that is actually hosting the iTunes library folder has to be running more or less all the time. Also, iTunes may not maintain all of its purchase authorizations between computers. If these things don't bother you, then this is definitely an option.