INTERVALS!
the best way to train cardio for martial arts is to train is as similar a fashion as the activity your are training for.
hrm, that's a bit of an awkward sentence.
take football (american) those big dudes that have to explode out and block the other big dudes, they arn't running miles and miles and miles, they are doing sprints and a lot of resistance training.
you need to do the same.do high resistance high intensity running/eliptical/rowing for the length of your rounds(or maybe a little bit longer, not a lot longer) and then take a rest (as long as your rests at aikido are).
running is not the best, something that offers resistance(like you would be getting in randori is better)
martialarts.stackexchange.com/users/21/Patricia
Crossfit tends to be great compliment to martial arts training. http://en.wikipedia.org/wiki/CrossFit
martialarts.stackexchange.com/users/8/Keith Nicholas
Sport-specific cardio
Unless you are quite good at a particular sport, the best way to improve your conditioning for that sport is to practice that sport at high intensity.
If you flag during judo randori, do more randori; if you get winded while boxing, then hit the bag more; if your aikido conditioning is poor, then go to aikido class more often, and make sure those classes are intense enough to be challenging.
In some sports, people can get so efficient at the skills involved that practice doesn't improve their cardio much.
This is particularly true in sports like Brazilian jiu-jitsu, where a black belt often doesn't have to expend much energy to fend off or tap a white or blue belt.
In these cases, using other forms of cardio to improve your conditioning can be productive .
Generalized cardio
I found it helpful to understand the three metabolic pathways.
Humans burn energy differently in (and recover differently from) different lengths of exertion: See this question on Fitness.
SE
for a more detailed explanation.
Intervals in any form are very productive for increasing your "wind" for multiple bouts of randori.
This includes sprints, Prowler pushes (PDF) , kettlebell work (swings, clean-and-jerks, or snatches) and a variety of other work.
The key is that allowing yourself to recover partially or totally between exertions allows you to train the explosive phosphagenic and hard-but-brief glycolytic pathways in addition to the steady-state, slow-and-plodding oxidative pathway.
You can also just do a single brief-and-hard workout on the order of 5 to ten minutes of all-out effort.
Glenn Pendlay has weighed in on the most effective conditioning for combat sport (which is tangentially related to aikido):
Remember, the goal is overall strength and condition, not to get good at any one particular thing.
Find 5 or 6 exercises that work for you and rotate through them, using one per workout.
Keep track of the reps you get in 10 minutes on each exercise, and try to improve.
Here are some good ones…
Push a prowler....
Kettlebell clean and jerks....
Farmers walk....
Take a barbell, a light one, and keep it moving without setting it down for 10 minutes....
martialarts.stackexchange.com/users/347/Dave Liepmann
There are two sources of energy in the body - aerobic and anaerobic.
Aerobic energy is the energy you get from the air you breathe.
Anaerobic is what your body can get out of itself.
Hint - aerobic energy is better!
It's easier to recycle (breathe out, breathe in, hey, new energy!) and takes less energy.
Using anaerobic energy releases lactic acid, and that's the part that hurts.
So, you want to develop better veinous/arterial circulation in the areas of your body where you need it -- along with good training to actually breathe in and out regularly without the breathing muscles getting tired or your body panicking from the high-intensity exercise.
Patricia's answer is quite good and, hopefully, this will give you some more.
martialarts.stackexchange.com/users/68/Anon
Kettlebells.
I wish I'd known about these things back when I was a lad.
The aerobic workout they deliver using an exercise technique like the Swing has to be experienced to be believed.
And they help with functional strength.
martialarts.stackexchange.com/users/54/Simon Peter Chappell
Tabata intervals/workouts are great for cardio.
I like doing them on a stationary bike, but using a timer you could do heavy bag drills too.
Toss in some strength training (basic barbell stuff) and you'll be golden.
http://www.rosstraining.com/articles/tabataintervals.html
martialarts.stackexchange.com/users/435/Wayne In Yak
The response regarding interval training hits the nail on the head.
It'll challenge your muscles and your heart, and avoid the loss of training effect that comes when muscles get used to a constant routine.
And it would seem to duplicate the exertions of a bout or fight, with activity highs and lows.
My training routine includes walking over mild, moderate and extreme hill sections in my neighborhood, with interval training on the mild and moderate grades.
The extreme hill areas (some are at a 10 degree or 1-in-6 grade) bring on breathlessness and muscle fatigue, and can only be done at a slow speed.
Walking up the extreme areas is like pushing a heavy stone on level ground.
I find that this cardio type training also helps build the muscle strength and balance needed for martial arts practice, stronger legs are good.
One advantage of hill walking is that going uphill utilizes different muscles in a different way compared to going downhill, so one gets the benefit of two different exercise motions.
Downhill seems more isometric, and research shows that uphill and downhill walking have different physiological benefits (one is better for blood sugar control, the other for blood lipids).
Running puts more stress on the muscles and joints than walking, which might be good or bad depending on the person.
I'am 63
so walking is my choice.
If one is going to be involved in a long contest, such as a 10 or 12 round boxing match, then fairly long distance running appears to help build the stamina needed to keep fighting over an extended period.
Some excellent boxers who tended to win short bouts and were not adequately conditioned were defeated by better conditioned and more wily opponents who were able to make the contest go the full number of rounds.
martialarts.stackexchange.com/users/1042/Larry B
So, I am looking to increase my general cardiovascular fitness to fit better into the martial art training I do -- Aikido but that should only matter a little.    The main question is  What exercises are worth doing to increase cardiovascular fitness for martial art training?  However, this is too generic and not martial art specific so, as clarifications, I am looking at increasing my recovery period between randori rounds/bouts and improving cardio more rapidly?  Please see  the meta question  as well if you feel this should be either downvoted or closed.
Rule 1: Cardio. What exercises are worth doing to increase cardiovascular fitness?
martialarts.stackexchange.com
<aikido><exercise>
martialarts.stackexchange.com/questions/219
martialarts.stackexchange.com/users/82/Sardathrion - against SE abuse
The best way to train cardio for martial arts is to train is as similar a fashion as the activity your are training for.  For example, football players do not run for miles and miles and miles - they do sprints and a lot of resistance training.  You need to do the same. Do high resistance high intensity running/elliptical/rowing for the length of your rounds nd then take a rest.  Running is not the best, something that offers resistance is better. CrossFit tends to be great compliment to martial arts training. Unless you are quite good at a particular sport, the best way to improve your conditioning for that sport is to practice that sport at high intensity. If you flag during judo randori, do more randori; if you get winded while boxing, then hit the bag more; if your aikido conditioning is poor, then go to aikido class more often, and make sure those classes are intense enough to be challenging.   Ways to increase cardio fitness for martial arts include  high intensity workouts, interval training like Tabata, crossfit, going to intense classes and resistance training suited to the length of your rounds. 
Some of the best ways to increase cardio fitness for martial arts include  high intensity workouts, interval training eg Tabata, crossfit and resistance training suited to the length of your rounds. Also going to intense Aikido classes. 