Yes, you can. Participating airline ONLY 'matters' for the trip from your Resort to the airport.
EVERY Disney Resort Guest is eligible to use DME in either or both directions - you just need to reserve in advance. By doing so at least the minimum required number of days, you allow Disney time to send you your DME packet including the yellow luggage tags. Simply attach one to each piece of luggage you're checking before you leave home; check that luggage at your home airport; and don't worry about it again until you see it in your Resort room!
Participating airline DOES matter for your trip home. Then, if you're not flying Song, Delta, Ted, United, American, Continental or JetBlue, you will have to bring your luggage to the DME bus - where the driver will stow it underneath. When you get to the airport, he or she will remove your and your fellow non-participating-airline-passengers luggage, where you'll have to bring it inside and check it yourself. Guests on the participating airlines will be able to check their luggage through from the Resort to their home airport.