Saiunkoku Monogatari's good too if you like Fushigi Yugi. It's pretty similar.
I recently just finished The Twelve Kingdoms so I'm trying to find more that matches up with this kind of theme. New world, historical, fantasy/slight political kind of anime. When I say political, I don't really mean straight up politics but like a "kingdom" where there is anarchy. I'm also going to recommend Kyou Kara Maoh since that was the first "kingdom" kind of anime I watched and loved.
Someone recommended Seirei no Moribito; that's an awesome pick! Female bodyguard, Balsa, protecting the prince while she had a troubled past with the royal family before already.. that's not really similar to twelve kingdoms, but the historical part matches up.
Well.. there's always Inuyasha and Clamp's Tsubasa series that fits the time/dimension travels..
Then there's also The Girl Who Leapt Through Time, a movie that's about.. well.. a girl who leapt thru time.. lol. she had many chances to go back in time. It's pretty cute actually, not that alike to 12kingdoms though. It's just a short cute and romantic movie, not series full of action and adventures.
Hakuoki is a good historical kind of anime if anyone's interested. It's about a girl who witnessed something she shouldn't have from the shinsengumi, some type of military group. They're trained samurais who are not as popular as real samurais with powers and enlisted under the shoguns or the influential figures. The series is all fun and historical...
But yeah, This is the end of my recommendations~