Kimochi in Japanese can be translated as feeling, especially good feeling. You'd usually say "kimochiiiiiiii" when you get in an onsen, or get a massage. Earlier when Sensei was trying to explain kankaku (physical feeling), I kept on trying to say kimochi, but he said no, it's different. Tonight however, we were working on a movement, and he said I need to do it with kimochi. Jibun no kokoro kara kimochi wo tsukau: Use feeling from my own heart. This seemed like a strange phrase to use with context, but it was indeed very important. Along with what we were doing, in Internal Chinese Martial Art lingo, I think he meant that I needed to root better. Relax your body, drop your weight, and move with kimochi. Another demonstration he had was with his hand/arm on mine. When he wanted to move it down, he didn't push, but he kimochi'd his arm making it heavy and weight down on mine.
This is something I've experienced in training with other teachers be it Tai Chi Chuan, Hawaiian Kenpo, or Boxing. It doesn't seem to be the mark of a particular style of martial arts, but a common component reflective of a matured practitioner.