Warm atmosphere with lots of slice of life moments truly relaxing and very touching.
The two main characters have a painful past and are orphans. They also grew up in host families.
However, 3-Gatsu No Lion focuses on shogi while Natsume Yuujinchou is rather a fantasy series with supernatural creatures (yokai)
Plotline and setting are different but I think if you enjoyed Natsume Yuujinchou, then you will enjoy 3gatsu no Lion, too.
They have a familiar feeling I cannot explain well...
-Main characters have similar background. Rei is an orphan who went through bad experiences and met nice people too.
-Both have heartwarming and sad/bittersweet moments.
-The narrative gives off a somewhat familiar feeling.
-Both are slice-of-life and you can see the struggles of MC in both of them.
Both anime have a slice of life genre, with heartwarming episodes. The stories surrounding the characters were simple, relaxing, funny and enjoyable, which either makes you smile or cry (out of the happiness) at the end. Totally recommendable to those who were feeling down and sad or stressed, this would totally make you smile and appreciate life.
Supernatural, light-hearted, charming, great characters, amazing soundtracks, there are just so many things that the two share together. If you enjoy Mushishi, you will definitely enjoy Natsume Yuujinchou.
If you want excellent character development, a flair of supernatural, and plot-driven tension that's a bit faster than the Natsume Yuujinchou series, PLEASE watch Boku dake ga Inai Machi (in English: Erased). There's a similar rural/psychological feel, and the protagonist also is a highly empathetic, responsible individual who is trying to protect people.