Aikido training does not have kicks. They have of course one-two techniques to avoid and lock kicks, but do not have any. They have strike techniques called "atemi", but did not use kicks in general, and they avoid them.
Do you know why is that?
You will rarely, if ever, find a martial art that truly uses kicks above the knee*, unless that art is centered around kicks. Most art are very concerned about their balance, and use kicks for disruption, not necessarily for damage. Aikido mostly uses the feet and legs for footwork. Kicks would just take time away from footwork.
By the same token, because you don't see it doesn't mean it isn't there. If there are kicks in aikido, they may just not show up in your average self-defense application, but they might be very useful, as explained in the other answers.
* [Uh-oh, I should specify that kicks during practice can be higher, of course, without any loss of face from the art.]
First, you have to understand what Aikido is, and what it is not.
Aikido is the final culmination of Ueshiba Morihei's training in:
All of these were important aspects in the development of Aikido, though the two largest influences were undoubtedly Daito-ryu Aiki-jujutsu (in which he trained the longest), and Omoto-kyo which preached harmony and world peace.
It's this latter element that led away from teachings of atemi; the later aikido becoming much more about a "way of life" and harmony with ones attacker. This is not to say by any means that it's a weak art (as my old aikido instructor used to say, "There's a reason we say harm-ony"). In this way, aikido became much more about the aiki techniques/approach of Daito-ryu, which clearly was more inline with the Omoto-kyo teachings, as well as the fostering relationship between Ueshiba and Deguchi Onisaburo (the second spiritual leader of Omoto-kyo).
Prior to this time, Aikido was known as Aikibudo, and was being taught to military personnel. At this time, there was a great deal of atemi, both punching and kicking. Still, however, it's important to note that this was derived from Daito-ryu Aiki-jujutsu most heavily, and much of the movement in Daito-ryu is derived from sword work (kenjutsu).
In (if I recall correctly) Morihei Ueshiba & Aikido: Aiki Budo, some of the early clips of Aikibudo showed kicking that I felt were similar to that done in Yagyu Shingan-ryu (and many other forms of jujutsu, of course – The human body can only move so many ways). There are many schools, also, that have gone back to an older, more aikibudo feel, and incorporate kicks in their modern training. This has more to do with the instructor and his background than the art specifically.
Often times, when you're questioning the absence of something, it's best to track back through the history of the development of the art; Ueshiba Morihei did not invent anything profoundly new, but rather, like aikido teaches, redirected the energy of his art in a new direction.
Not sure if I understand this right: Your asking why Aikido has no kick attacks and not why Aikido has no techniques against kicks, correct?
Aikido, as it is today (at least the Aikido I have seen so far), has no attacks at all. It's principle is avoiding conflict on all levels. Attacking means conflict. Even the defensive techniques of Aikido are not about disabling the opponent. Instead they focus on harmonizing your energy with the attacking force and transforming the attack in something else. So praticed right there will be no fight and no opponent.
This is why Aikido has no attacks in any form.
Aikido does not expressively have attacking kicks. Several techniques do open uke to receive a kick but those kicks are not practices. It is assumed that the practitioner knows how to kick from a different art or uses a knee strike, or punches or does not bother with the strike and does the throw/pin directly.
Aikido does have a wide range of techniques that can be applied against a kick. Generally speaking, any kick is best avoided and not soaked.
Aikido aims to protect yourself from injury, as well as your attacker (if possible). It aims to use very little, if any, force by itself, but instead redirect the attacker's force back onto the attacker.
A kick is pretty much the opposite. A kick intends to apply a great amount of force onto the target, trying to directly damage it, by force.
Stokman Sensei covers defenses against kicks in his book. I haven't consulted it in several months, but if I recall correctly, he was also puzzled.
As a young student I was told a teaching story - that part of the genesis of aikijutsu was when a Samurai lost his sword on the battlefield and had to take the sword away from an opponent. Most of the kicking and punching arts were limited by the opponents armor, but joint locks worked just fine. I doubt that has any authenticity as other than a legend, but I like the concept.
I believe their are historical reasons for the lack of kicks, most of which have been mentioned in other answers including:
So the question that seems to beg to be asked is: why hasn't aikido adopted kicking in all the years and all the styles?
I would propose the real answer is that any kick above about the ankles is far, far too easy to counter, and the counters are very, very devastating.
High kicks are the realm of sport styles with rules preventing the devestating counters that are possible to a well trained martial artist.
No one in their right mind (okay, why are we martial artists, then, but that is another question) would want to take the ukemi from kick defense practice!
Simply put: because the mat isn't that pretty. In my experience in Aikido, lifting the leg high will only set yourself up for a leg sweep or joint lock. When you are allowed to grab your opponent's kicking leg and sweep the knee, most people won't risk it. It's a good way to end your time in the ring for good.