Weight training is fine for isolation of specific muscles. Running is good for cardiovascular conditioning.
Specificity of training might require a closer look. I have found weapons work has benefitted both my grip and my upper body. Certain yoga positions has helped the stability of my base of support. Swimming has benefitted me both cardiovascularly and helped condition me for turning-on-center movements. The twisting motion in water with the resistance increasing as I try to swim faster is a good CV workout as well. Playing an aiki version of tag (chasing by doing forward rolls, backward rolls, etc.) with children is a fantastic vestibular conditioning tool. It keeps you from getting too much vertigo info from inner ear, and helps ukemi.
As for those who believe that only aikido will help aikido, you may want to consider that the number of micro-tears (musculo-tendonal or other) which occur with training can be reduced if the body is given a rest, yet one can rest and still train if there is variety in training.