dgd hates me

She doesn't hate you, she is trying it on, even at that young age.

You are doing great, sound like a lovely gran.
Dgd is pushing boundaries and finding a reaction.
I also would ignore what she is saying, not try to talk to her about it.
Don't give a reaction.
My loving dgd once told her mum that I'd smacked her, insisted upon it. I said nothing to her about it. Eventually one day mum said that dgd had something to say to me..an apology.

I have had a couple of sleepless nights over this but I know I am the gran who does things, I let her paint (to heck with the carpet its old anyway), cook (the floor will wash), feed the cats (I know I will have to wash her hands after), jump in the puddles or plodge in the river (her clothes will clean) etc. She doesn't need lots of new toys/clothes/dvds etc.
Being with her has helped get over my cancer as I just cant worry about me when shes there. I just cant help being upset that she says she hates me and pulls her hand out of mine.

ethelbags your DGD is three years old and children of that age say all sorts of things that they have no idea the meaning of, she sounds like she's just repeating something she's heard without even knowing what it means. You should really ignore what the other grandparents do or don't do and concentrate yourself on making a good relationship with your DGD , there are lots of things you can do that don't cost anything and that aren't unhealthy. Take her to the park for fun or let her help you cook or just play games and read to her and ignore any remarks she makes as the more you make of it the more she will do it just to get a reaction she is testing her boundaries with you as children do as they grow and realise how to get the power , so ignoring her is the best thing to do.
I hope things get better with time as there is nothing like time with ones DGC.

And don't try and convince her that you love her very much. Just BE loving.

Don't try and convince her - don't talk about it - ignore any silly behaviour like hiding behind sofa.

You are the adult and she is too young to be embroiled in your feelings of rejection.

Just be a jolly gran according to your rules (e.g. no need to stuff her with sweets as you don't agree with it) - laugh off any nonsense and distract her.

Rise above it!