A good way to allow your shoes to dry faster is to pull the insoles out of the shoe when you are not wearing them. See the picture to the left for an example of how I store my insoles. I tend to pull the insoles out so that just the ball of the foot is sticking inside of the shoe; that keeps the appropriate insoles with the correct shoes to make it faster for me to put the shoe back together before wearing it. This will allow air to circulate within the shoe, which will speed up drying time. This only takes me a few moments at the end of a run and a few moments before going out for a run to accomplish, and can really speed up the safe drying of your shoes by quite a bit.
Another way to speed up the drying of your shoes is to get them off of the floor. If you put your shoes on a rack, more air can get under the shoes and you can allow your wet shoes to drip onto the floor rather than having any water that might drip staying in contact with the fabric of your shoes. I recommend putting the wetter shoes closer to the floor than your dry shoes, so that you do not drip into your dry shoes. As you wear the dry shoes out in the rain and puddles, you can move your previously wet shoes higher up the rack to replace with your newly wet shoes.