My puppy is about 4 months old part yorkie and shizu (sp)and some other breed we are not sure of and his coat is very greasy and he has a bad odor. What causes this and what can I do about it?

Check the fat content of his food and lower it. A good vet/pet store brand of kibble with a small amount of canned food or fresh meat, not some cheap grocery store brand.
Give him a gentle bath, with just a tiny amount of soap and a vinegar water rinse to clear out any residue.
Too much bathing and dry skin is not good either.
Smell inside his ears. Yeast infection, bacterial infection, and mites cause an awful smell. Take him to the vet if they smell or have brown stuff in them. You will be taught how to clean them as well as treat them.
Is it winter where you are? Is your house overheated and dry? Do you have wall to wall carpeting?
He could be reacting to any of those. Make sure he has fresh air every day.

Sounds like allergies. A lot of dogs get smelly or greasy feeling when something in their food isn't agreeing with them. THe most common allergen is wheat. If your feeding a commercial brand with a lot of fillers, that could be the reason.

I would try adding some omega oils into the diet. Cod liver oil, salmon oil, even a can of sardines in his diet would help heaps. You can also try using a humidifier to put some extra moisture into the air.

If you are thinking allergies don't bathe him with commercial soaps. Owen has allergies and we have to use a medicated shampoo for him and bathe him twice a week. I would suggest a vet consult just to see what it is so you can treat your baby properly.