Look for a local area company, such as Unwanted or 1-800-GOT-JUNK?, that hauls away appliances for recycling. Some companies will do this for free, while other charge a fee that varies with the item and location.
Place an ad on Craigslist for removal of the appliances. Some people may be interested in stripping parts from the appliances, or they may just want the appliances and will haul them away from your home for free.
Purchase a new replacement appliance. Most appliance stores, like Lowes, will remove your old appliances and haul them away when they deliver the new appliances. Inquire about the policy before making the purchase.
Ask around or place an ad online and pay someone to come and haul away your appliances if you can't make that happen any other way. People from your community or church or online may be willing to haul the appliances away for you if you pay them a small hauling fee, such as for the cost of gas and for their time.