Media Center determines what tracks are in an album by using the field/tag [Album Artist]. The tracks with the same [Album Artist] and [Album] tags become an album. Typically, those tracks are stored in the same folder. For compilation albums, [Album Artist] is often set to [Various]. Albums are typically stored in a folder [Artist]\[Album] which does not work for compilation albums, as you note. If you want to collect all the tracks from a compilation album into a single folder, you need to come up with a name scheme for that. You can use [Various]\[Album] or just drop the artist part and use the album name, hoping you do not have 2 compilation albums with the same album name. If you set [Album Artist] to [Various] you could use [Album Artist]\[Album] and all the compilation albums will be in the Various folder.
Media Center has a very powerful tool known as Rename, Move & Copy (RMC). It allows you to do those actions based on tag values. It not only moves the files, but updates the library to the new location. See the Wiki for a full description. For example, if you have the tracks for a compilation album selected, you could Rename these tracks to a new folder using a Filename rule like c:\Music Files\[Album] The tool will show you the original file name and the new file name, so you can be sure you did it right.
You should try the tool on a few files, like one album, before doing a wholesale change. You can really screw things up with this tool if you are not careful. But, used correctly it is incredible powerful. You should also set the [Album Artist] to [Various] for the compilation albums.
To get to RMC, select the tracks of a compilation album, highlight them, right click and select Library Tools and Rename, Move & Copy. Look at the preview before committing. Select Rename at the top and fill in a Filename rule.
Take it slow but once you master it the changes will be easy.