There are Native American characters in modern comics by DC, with a few of them being:
● Sarah Rainmaker (seen in the Wildstorm Universe, as well as the New 52’s “The Movement”)
● Manitou Raven, first seen in JLA #66
● Manitou Dawn, first seen in JLA #75
● Miiyahbin Marten, aka Equinox, in Justice League United
As far as diverse teams from the modern era go, The Others had a decent amount of variety, including:
● Sky Alchesay (another Native American, btw)
● Kahina/The Seer
● Sayeh (Kahina’s sister)
● Ya’Wara
Another modern era team that was diverse is The Movement. Among their ranks were lesbian (Virtue/Holly Ann Fields, and Rainmaker), gay (Burden/Christopher, who was raised Amish) and asexual (Tremor/Roshanna Chatterji) characters.
For modern Asian representation (and in addition to the main characters from New Super-Man) there’s the Tai Pham Green Lantern from the Green Lantern: Legacy OGN (apologies if someone already mentioned him; if so, I missed the mention).
While not an Asian character created in the modern era, Katana has been prominently featured in modern comics, movies and TV.
A more recent Asian American addition to the fold is Chien Na Wei, aka China White, who debuted in Green Arrow: Year One.
For additional African-American representation, there’s (among many others):
● Steel (star of his own ongoing for several years, as well as appearing in other media)
● Natasha Irons
● Dr. Shay Veritas
● Ron Troupe
● Milestone characters (Static, Hardware and Rocket, among others; they have also endured for years and have been featured in other media)
● Bloodsport
● Tangent Earth Superman
● Earth 23 Superman
The Blackhawks feature a wide array of diversity, especially the Post-Crisis team (with Asian, African-American, Mexican and Malaysian members) and the New 52 group (featuring Asian, Hungarian and Ukrainian members).
Based on other folks’ observations, as well as the above, the DC Universe seems fairly diverse to me, especially with regard to black characters.
Every character in this entire thread may not be a household name, but their validity isn’t measured by how well-known or unknown they are. What counts the most is that they exist, and that they have all been important contributions to the DC Universe.