The first thing that you need to do on your way to using chewing gum as air freshener, is to go on out of your house and actually buy a lot of gum. Now the good thing about this, is the fact that you will get to handpick the gum that you like, and the ones that you feel smell rather good.
This is not always the case with air fresheners, so you already have the upper hand on things, as they don't tend to be custom made to your liking, but are instead made for the mass market instead.
Now once you have the gum, what you need to do, is to break them open, since all the flavour and smell is stored inside the gum. Breaking them open will help release this smell and will make it smell better.
Next you need to take this broken gum and place it in the air vents in your house. This is going to see the wind hit the gum, and then go on and distribute its smell across the house.
Alternatively, you can also go on and place the gum in the air conditioners vents, which would end up producing a very similar result and will help you get the minty smell of chewing gum flowing through your home.
You can experiment with different flavoured gum, to see how they will smell when you use them.