Truly successful logos should work in black-and-white or grayscale. You should be able to run the logo through a crappy fax machine, and the logo design should hold up, that is, it should be recognizable and readable.
Good logos should also be able to be scaled down to a thumbnail size, and should still be recognizable. When working on your logo designs, scale them down, or zoom out, and see if you can still read them.
When submitting the logo assignment, you must also include black-and-white or grayscale versions of the logo options. If you have background art that conflicts with it, submit the B/W versions without the background art.