Do I have legal rights to my kids after their mother passed away?

In Arizona, as a biological parent you have priority for being the primary residential parent, if the other parent is deceased. There are exceptions, so what I suggest that you do is consult with an experienced family law attorney in your area to discuss your options.

A biological parent should have priority over a grandparent.

Unless she is the legal guardian appointed by court, you are the one entitled to custody. As for no hassle, that is up to her. My guess is that she will raise a big hassle. If she won't give you custody, go to the police and if they won't help, then file a petition to get the children from her.

I assume that you and the mother of your children were unmarried, and I assume that the deceased mother's mother (the maternal grandmother) has taken these children in following her daughter's death. If the grandmother does not have an order of the court granting her custody of the children by way of adoption or through a guardianship, then your claim to custody of the children is superior to that of the grandmother. That means, legally, that you can simply pick up the kids and take them home with you. They are your children. You have a right to custody of your children when the other parent is dead. My guess is, however, that the maternal grandmother is not eager to hand these children over you. If she is not the adoptive parent now and not a guardian of the children, yet she won't hand the children over to you, you could try to call the police and ask them to assist you in retrieving your children from the maternal grandmother. Technically, she is a kidnapper if she will not voluntarily relinquish the children to you. My experience with the police, however, has been that they couldn't care less about enforcing parental rights in this manner. So that leaves you with having to go to court. You will likely have to file a petition to establish by court order your paternity of these children, and to seek custody of them as their biological father.

You are the proper custodian of your children. Get an atty to help with the transition to create as little drama for the children as possible.

If your parental rights have not been terminated or otherwise restricted, you have a superior right to the children than the parents of your deceased former spouse. that said, barging in and demanding the children would be difficult, perhaps even scary, for the children, so proceed accordingly.