Classical martial arts are mostly moving lots of air, I can't remember the last time I saw someone in the octagon doing some Karate kicks and shit, MMA for the win and for real life protection.
David Loiseau and GSP use the spinning back (horse)kick - Kyokushinkai.
Andy Hug (K1)- Axe kick - Kyokushinkai
Other Kyokushinkai/MMA fighters
Bas Rutten -
Francisco Filho
Obviously they rely on other techniques drawn from other styles to build a complete fighter.You're not going to see the old fashioned karate chop utilized very often if ever, but some of the lower body kicks can be devastating as Muay Thai if used properly.
Jai-White is a kyokushinkai btw