Marge, I'm sorry this is happening but I'm so glad you asked. The "grease" stains are being caused by your liquid fabric softener and/or dryer sheets leaving deposits on your laundry. Sometimes if using a liquid softener, the spots will appear blue.
Try rubbing the spots with a wet bar of soap or liquid dish washing soap and then rewash. Don't use laundry detergent because it can actually set in the stains.
To avoid the spots, place the dryer sheet on the top of the clothes and start the dryer immediately. Don't overload the dryer and don't bury the sheet in the middle of the load. Dryer sheets should not be used when you are using air fluff or low heat.
For liquid softener, dilute with an equal amount of warm water before adding to the washer to help disperse the liquid. If your machine has an automatic dispenser, clean it out regularly by flushing with hot, hot water. You'll be amazed at the clumps that will fall out. It's those clumps that stain your laundry!
Specific questions? Just ask here.