The Royal Diaries and the Dear America series were the first ones I thought about when I read your post, but they aren't series with recurring characters.
The Birchbark House by Louise Erdrich tells about the same time period as Little House, but from a Native American perspective. There is death and hardship portrayed in the plot. There are sequels too.
Anne of Green Gables and Emily of New Moon and the other series by L.M. Montgomery might suit, but there are dramatic, tragic events. I still can't read about Matthew's death near the end of Green Gables or about Walter or Dog Monday in Rilla of Ingleside without getting teary-eyed.
They are not a series and the reading level is higher than the Little House books, but Karen Cushman has written several books about Medieval times that are terrific. They feature spunky, smart girls and she's done a lot of research for historical accuracy
- The Midwife's Apprentice
- Catherine, Called Birdy
- Matilda Bone
I feel like I am missing something obvious, but I can't think of what it could be. If I do, I'll post again, lol!