My daughter was a tornado, too, at that age. However, I now have a teen and a tween and I JUST had this frustrating conversation with them, yesterday - "I am more than just a cleaning lady!". ARrrrrrgggghh. it's not just a 3-year old concern! But you must stay on them.
Totally organize her room. Work with her on it. Take out what is not needed and give it away if she will never use it. Toss the little stuff. Get her used to organizing it. If she insists on "saving it", make the suggestion to "move it to another part of the home", temporarily, and box it. If it sits "temporarily" untouched for over a month - donate it.
Limit the amount of stuff in her room. If she has an excessive amount of toys, leave some, box up the rest and you can relocate them, to storage and rotate them out every month or two.
Look through her clothing - does she need that many clothes? If you have her winter and summer clothes in her room, put the opposite season in a bin and take out what is needed for this season.
Make organizing easy for her - use easy to store bins that possibly slide or sit on a shelf. Label the bin and color code, if necessary.
Get her in the habit of doing small chores: When she wakes up, she should make her bed, open her blind/curtain, put all dirty clothes in hamper, kiss her stuffed animals and give them a bed to lay in. Whatever the routine - just strive for consistency. She's still young and when she makes a huge mess, it could overwhelm her too.
After you have done all of this, set up a system to "check on her room". At the beginning, make it every other day. Praise her when you see her bed made or anything good that she has done. After that, maybe on a weekly basis, she may enjoy a walk with you, bike ride, trip to the park, ice cream cone or just a "bedtime story" in her room... whatever she would think would be a great reward for being "responsible". Every couple weeks, you may have to help her do more of the nitty-gritty stuff to keep order.
Independence can mean two outfits for a day - no more. You won't stifle her. She can wear her third choice the next day.
Always make sure before the holidays (November) that all toys and stuff have been completely sifted through. Get her into the habit of working with her to donate what she doesn't need so that she becomes aware of others who are in need that she can help.
My children still enjoy it when I help them clean their rooms, every few months. It's also good to be able to have access to your child's room, especially when they are older and trust you. That's why you start when they are young. it's a great bonding time for them (if you can look beyond the mess) to feel that they have your undivided attention and may be apt to open up about private topics.
Good luck.