You could try to fill it with paper (e.g. newspaper), then you could put them in front of blower.
If the paper gets wet, replace it (or use the fan if you have one).
lifehacks.stackexchange.com/users/21/nicael
Whenever I attend music festivals I end up using the hair dryer in the hotel room.
Not as easy as a clothes dryer, but it works :)
Note: do not leave the hair dryer in the shoes or they'll light on fire.
lifehacks.stackexchange.com/users/1310/braden
Throwing them in the dryer is obviously the quickest and easiest way (assuming you have one), but as you ruled that out due to the fact that you don't want to dirty your drier up, you'll just have to simulate the experience.
First off, loosen the laces a bit and pull the tongue out to expose the interior of the shoe as best you can.
From there, you have a few options, but they pretty much all boil down to "apply air and/or heat".
If it's winter and you have the heater running, or its summer
and you have the air conditioning blowing, place them upside down on top of the heating/cooling vent.
If it's a particularly sunny day, put them out on some concrete or blacktop.
If all else fails, put them in front of a fan.
lifehacks.stackexchange.com/users/168/Sterno
Even if you would rather not put your dirty sneakers in the dryer directly, you can still use the dryer to dry them... Locate your exterior dryer vent and place your shoes under the vent.
The hot air blowing from the vent will speed up the drying process.
Or if your dryer vent is on the roof or otherwise inconveniently located... Place your shoes inside an old pillow case and tie the pillow case shut.
Then toss the bundle in your dryer and tumble dry on the lowest setting.
Provided that your not trying to dry huge heavy boots
the dryer should be not damaged by the shoes bouncing around inside and the pillowcase should contain any dirt
the comes loose from the shoes.
I use option 1 for my combats and option 2 for my chucks , both work pretty well.
lifehacks.stackexchange.com/users/32/apaul
You can use a hair dryer but only leave it in the shoes for a couple minutes because it may warp the shoes, also if the dryer has a cold setting that works best!
lifehacks.stackexchange.com/users/27937/E_lucille
A high volume leaf blower works well.
Just stick the output nozzle inside the shoe and turn the machine on high.
I have used this method on soaked work boots and it works great!
lifehacks.stackexchange.com/users/19061/user19061
If they are really wet then cat litter or rice absorbs moisture quickly, but then you have to clean that up too.
In Japan several electrical manufacturers make portable dryers, with attachments for shoes.
Many also have an ion generator function to remove odours.
For some reason they only seem to be sold in Japan.
lifehacks.stackexchange.com/users/14058/ａａａ
If you're able to get the shoes 95% dry (following the suggestions in other answers), you can get them the rest of the way dry by spraying the shoes with a drying agent like 99% isopropyl alcohol.
They'll be wet, but the alcohol and water form a mixture which will evaporate more easily than water alone.
(It will work whether or not an azeotrope is formed.)
A small fan blowing on the shoes will aid the effect.
It doesn't need high airflow.
Edit: A previous version of this answer indicated acetone could be an effective drying agent (if you're sure it won't weaken the rubber or fabric), but wikipedia indicates that doesn't work: Despite its common use as a supposed drying agent, it is not effective except by bulk displacement and dilution.
lifehacks.stackexchange.com/users/4092/piojo
Secure the sneakers to the lid of your car's trunk (boot for you English speakers).
Duct tape works well for this purpose.
Drive down the road at highway speeds.
As long as it isn't raining or snowing, the air flow off the roof of your vehicle will remove the moisture very quickly.
Note: the more humid the air, the longer this process can take.
Note: using a non-paved road will definitely kick up dust which may make the sneakers muddy.
You may need to look at after using this procedure:
Removing duct tape from my car without paint damage?
lifehacks.stackexchange.com/users/10/Adam Zuckerman
If you have an electric water heater, you can sit them on top of it overnight and they will be dry in the morning.
lifehacks.stackexchange.com/users/21144/Janice
Use a dry dry off towel put the dry towel in the shoes and it will soak some of the water out of the shoes
lifehacks.stackexchange.com/users/24128/Drell
I stepped in a deep puddle and now both of my sneakers are thoroughly wet. I've tried opening them up to let them air dry, but this takes far too long. I'm also afraid to put them into the clothes dryer (for the sake of both the pair of shoes and the dryer itself)  How can I dry them out quickly to wear again?  (Sneakers = trainers for you English)
How to quickly dry wet sneakers?
lifehacks.stackexchange.com
<cleaning><clothing><water><shoes>
lifehacks.stackexchange.com/questions/228
lifehacks.stackexchange.com/users/15/Mooseman
One effective way to dry wet sneakers is to fill them with paper and place them in front of a blower. Another trick would be to use a hairdryer or simply leave them outside on a sunny day. Other possible methods include blowing them with a leaf blower or using rice or kitty litter to absorb the moisture. The options include stuffing with paper and placing in front of a blower, using a hair dryer, loosening the laces and putting outdoors on a sunny day, placing next to the exit vent on the dryer, using a leaf blower and duct taping to the trunk of a car and driving at speed.
Stuff them with paper and place in front of a blower. Use a hair dryer, but do not put in the shoes or they'll catch fire. Loosen the laces, pull out the tongue, apply air and/or heat, e.g. outdoors on a sunny day. Put the shoes next to the exit vent of the dryer where the warm air comes out, or put them in a pillow case and then in the dryer at lowest setting. Use a leaf blower - place the nozzle in the shoe and switch on. You can use cat litter or rice to absorb the moisture - altho you then have to clean it up. If in Japan you can buy a shoe dryer. You can spray a drying agent like alcohol onto the shoes to speed up the drying at the end. Stick the sneakers to your car boot using duct tape and drive down the road at high speed. If you have an electric water heater place them on this overnight. Put a dry towel into the shoes to absorb the moisture.