If you look through old requests (by topic--see column of topics) you will see many requests about babies and toddlers that have sleep issues. You are not alone. = )
It sounds like the whole family must be exhausted, not just the 3yo.
She could be having bad dreams, or there could be strange sounds in her room (I mention this because my husband tried laying in our son's bed at night and found that he was right, it did sound "creepy" in there, because sound was bouncing down the long hallway), or who knows. It will probably correct itself as she gets older. But for now, to let everyone get more sleep, maybe you could lie down with her, or stay the night with her once or twice (if the bed isn't big enough, maybe on the floor next to her). If you are with her at night, you may realize what is causing her to wake up (dreams, sounds), or if she just isn't getting to sleep in the first place---and you can work from there.