This isn't an uncommon problem, and one reason I wanted to reprint your question is because it might encourage someone who is considering remarriage to sit down with their children and listen to their concerns in advance.
From what I gather you didn't do this which is why I urge you to sit down privately with each of them and listen to their worries. These may range all the way from financial concerns ("This new wife of his is going to steal our inheritance") to emotional reactions ("How can he turn his back on our late beloved mother? No one will ever take her place in our hearts"). Rightly or wrongly, they are probably thinking more about how your remarriage impacts them than what it means to you.
As you listen to them, try to put yourself in their shoes and understand their concerns. Don't argue with them or be defensive; remember the Bible's admonition: "A gentle answer turns away wrath, but a harsh word stirs up anger" (Proverbs 15:1). At the same time, let them know you are thankful for your new spouse, and you hope they can get to know each other in the future.
Most of all, pray for your children. Losing a parent has been hard on them, and it takes time to heal. Let them know you still love them and will always be there for them.