What do I do with the neutral wire when there isn't a place for it?
Electric dryers are typically designed to be protected by 30 ampere breakers, so I'm not sure why you would have a 50 ampere circuit. You'll want to check with the manufacturer to be sure what size overcurrent protection is required. If 30 ampere protection is required, you can simply swap the 50 amp breaker for a 30. Since the conductors are 8 AWG copper, there's no problem with using a 30 amp breaker instead of a 50.

A 3-prong dryer cord typically uses a NEMA 10-30 plug, which has two "hot" and one neutral prong. So I'm not sure why your setup wouldn't have a neutral.



3-prong dryer cords do not have a grounding prong. When using a 3-prong cord, the dryers chassis is bonded to the neutral via a bonding strip/wire. There should be instructions for this in the manufacturer's documentation, or on a label on the unit. 

If you're running all new wire for this circuit, I'd recommend that you (or you may be required to) install four conductors instead of only three. This will allow you to have two "hot", a neutral, and a grounding conductor. You'll then follow the manufacturer's installation instructions for a 4-prong cord.