The style of swordsmanship I study is Muso Jikiden Eishin-ryu Iaijutsu, more specifically, the Yamauchi-Ha branch under Sekiguchi Komei Sensei. At the dojo we study all sorts of skills in addition to regular iai training, from empty hand iai kata applications to saya strikes. This style is based from the techniques of the Tosa clan.
At this dojo we focus on practical techniques as well as kata. Iai is the study of drawing the sword, but at the Komei Jyuku we study more than just that. It is amazing to see how every piece of the sword can be used as a weapon, even the sageo. A lot of this is hard to explain. It is best to find a dojo and see what you can learn.
At the Lubbock dojo, Bushey-sensei teaches us with very creative means. While most training time is spent on iai kata and kumitachi drills, there are some days when we do unique exercises devised by Bushey sensei. An example of this is "the forest", where we use shinai to attack imaginary attackers (punching bags) placed in various positions. Then there is "the gauntlet" where we form two lines of attackers, with one defender in the center. One attacker can attack the defender, and the defender has to defend and kill the attacker and move to the next guy, until everyone is defeated. These types of exercises build skill and timing as well as give a nice change of pace to the repetitive training necessary to iai.
If you are in the Lubbock area, check out the Texas Komei Jyuku. I am usually there, so just stop by and see what is going on. Bushey sensei will be glad to answer any questions you have.