My girls used to be like that. They are still all about dh though. Some days they still are like that too.
Our routine in the morning is they get up with daddy, catch up and we all have brekky together. We all get ready for the day. Most days we follow Dh to work, then we go our separate ways. I think them knowing where is he going helps them understand that he will be coming back at some point (some days we surprise them by taking them to Dh's work so they can show him stuff, or havea milkshake or whatever). If we stay home, then they walk him to the car, and 'help' him get his work stuff in the car. Then they wave out the front, make as much noise as they like saying goodbye, and being silly.
The biggest thing that we do is Dh will call them around the same time every day. Before we say our goodbyes, dh will tell them he will call after playschool or whatever thing we are doing. That will normally stop any tantrums.

FWIW, when I was little my dad had to sneak out of the house to go out. The same happened with my mum and her dad, he would have to sneak out too. But instead of dh sneaking out we are trying to ease their fears through reassurance. It works really well for us.