I'm assuming that there is more to your story as to why your 3 yr. old granddaughter is with you which is also adding to these defiant times.
Remember ~ Every child is different. For my son, I say "Do you need a time out" and he's done. On the rare occassions that does not work I take him to time out for 1 minute (I chose one minute because I still have to hold him in time out and he sobs through most of it so I feel he does not need anymore than that). Have a designated spot in each room so that when she gets in trouble you immediately take her to the time out spot. You may have to stay with her ~ As I said, I have to stay with my son and hold him so he stays.
Your grand-daughter is 3 years old and is still trying to figure out how to communicate her feelings. You may try talking to her using aides such as paper plate puppets that you have put a smiley face and sad face on ~ You can give this to her and ask her which one she feels like, then why. Be open to her answer and remember at this age, the smallest things seem like the world is ending to them.
I think aides help tremendously. How about making a rule chart on poster board? Take the things she mostly gets in trouble for and put them on the board ~ "Listen" use a picture of an ear; "Clean up Toys" use a picture of toys; and so forth.
Pick your battles ~ Not every battle is worth fighting right now.
REWARDS!!!! MAKE SURE TO REWARD HER WHEN SHE HAS DONE SOMETHING GOOD OR RIGHT. You want her to see what it's like when she is 'good' or has done something nice. I'm not a big sugar fan for kids so I recommend small toys (wrap them and keep them hidden for these occassions), Popsicles made with Splenda (Walmart in a small pink box), use her interest to help guide you on what to use and look at your local dollar store.
Have you thought about speaking with a child phycologist? They have the tools and resources to really draw out what your grand daughter is feeling and why she is defiant. This would be my strongest recommendation ~ Don't wait because it will only get worse with age.
My son is very stubborn as well ~ just not when it comes to doing as he is told. The issue is really with her being defiant, unfortunately her stubborn ways add to that. I look at them being stubborn as a blessing for later in life ~ It means they will never settle for what life Gives them, they will always strive for more; They will have a target and goal and won't give up until they get it; They will be successful!