Since I have extremely thick hair it's a little difficult for me to get a "quick" blowout. If I'm pressed for time, I will typically grab the hair dryer and start drying my hair while using my hand as a brush (yes this does create a bit of frizz). After doing that for about 5 minutes, I will then separate my hair into sections and style with a big round brush. 

On the other hand if I want a quick blowout but have a little more time to spare, I will let my hair air dry out of the shower for about 20 minutes. I prefer doing this because I won't have to put as much heat on my hair while I'm drying and hopefully minimize damage and split ends. I will then separate my hair and dry each section with the round brush. 

I also use an ionic hair dryer which speeds up the process! 

Hope this helps :)

I try to let my hair dry about half way. Typically I do my makeup and get dressed during that time. Then I apply heat styling cream all over my hair.  I grab my blow dryer and begin to dry my hair and brush my fingers through it for a few minutes. Then I use my round boar's hair brush to finish everything off in a way that looks put together, but not super stick straight. Once I'm happy with the results I add a tiny amount of the heat styling cream over my dry hair to keep the frizzies at bay. It takes me about 20-25 minutes to do this. Not quick, but i have medium/long thick hair so that is pretty good for me.

Fortunately though it's summer time so my natural beachy waves are in (and I don't have to blow out my hair as much).

I  comb my hair wet and blow dry it turning my head upside down and blow dry it.

The key is to use the concentrator attachment for the blow dryer. It's a world of a difference when you do that. I let my hair dry until it's damp-ish but NOT starting to go into waves yet (very fine line, haha) and then just use a normal brush with the concentrator. It looks almost as good as using the straightener but I'll often go wuickly over the top with the straightner to give it a more polished look.