On Mac, if the traveling drive has the same name as the drive on your other computer LR will most likely find the photos automatically.
If for some reason you can't or don't want to name the drives the same, relinking photos in LR will be much easier if all your image files are contained in subfolders underneath one top-level parent folder. At least then you just would need to relink that top folder.
On Windows this is a bit more problematic; as flash mentioned drive letters come into play, and become part of the file path. You would need to assign the same drive letters to both drives, which in itself is also problematic because you can't have two attached drives with the same letter. Maybe a Windows expoert can explain how to accomplish this.
And as Francois mentioned, often using the Export/Import from Catalog is better, especially for smaller sub-sets of your master catalog.