Maxima's father's family seems to be of Basque origin, but supposedly claims indigenous ancestry as well (although the claim is indigenous royal ancestry, so it seems far-fetched). Still, that ancestry would be as far back as Queen Silvia of Sweden's or Grand Duchess Maria Teresa of Luxembourg's.
As for Asians, I agree with whoever earlier mentioned interest in an Asian/European royal-on-royal marriage. I'd love to see the two thousand year pedigree of Japan marry into one of the ruling families of Europe and bridge the East/West divide in monarchies. And it's an obvious one, but I feel compelled to mention the former Princess Alexandra of Denmark (now Countess of Fredricksborg), who is of mixed Chinese and European ancestry.
I wish there was more media attention for Princess Angela of Liechtenstein (born Angela Brown), whose son is the highest-ranking (in terms of succession) person of significant African ancestry in Europe. I love the pictures of the young Prince Alfons von und zu Liechtenstein in the Liechtenstein national dress. It's the perfect illustration of modern multiracial identity.
Also, Noor of Jordan's marriage wasn't completely well-received, even within the family. She was famously referred to as Hussein's "CIA wife" by some of his children early on in the marriage, demonstrating a distrust of her American origin. Anyway, Noor (born Lisa Najeeb Halaby) was an ethnic Arab of Syrian, English, and Swedish American descent. (She is an Arab because her patrilineal ancestry is Arab).
However, recent events in Jordan have seen the BELOVED (seriously--ever met a Jordanian who doesn't gush about how beautiful Rania is?) Queen Rania publicly protested against by ultra-conservative tribal groups for her Palestinian origins. It's actually the first I've ever really seen a split in the Palestinian/Jordanian bond discussed so openly.
One of the Princes of Saudi Arabia just married interracially (if European and Arab are actually different "races," but we'll say that for the purpose of discussion) to one of the nieces of the Duke of Northumberland (actually I think she is the current duke's cousin).