Neutral:
* Mixed breed
* Crossbreed
* Hybrid
* Cur
* Dog of mixed ancestry
Informal:
* Heinz 57
* Mutt
* Doodle
* Poochie
Formal:
* Canine of mixed lineage
* Mixed-race dog
* Composite breed
Literary:
* Composite
* Mélange
* Hybrid
The best word to use will depend on your specific needs and the context of your writing.