I have the problem of having a 3 prong outlet, and a four prong dryer cord. I bought a three wire cord to replace it and I understand that I have to hook up the black and red and the white is neutral. Is there any way to keep the grounding "on/in" the dryer rather than externally grounding it?

beckyn - you said that you have a three prong wall receptical and a three prong new cord :: Ok there should be no worry <because the wall receptical is wored with {should be } 10/2 w/ g ; the wall wire is one white it is a hot wire and a black wire and it is a hot wire and a bare copper that is the ground wire:: The new threeprong is the same :: {no problem there} on the back of the dryer <you have a electrical terminal block <and it has three
Screws for the new cable to hook up tyo :: The two outside screws are for the hot wires <and the center screw is for the ground wire to hook to :: You ,do not want to use the four prong cord!! {you want to goosd luck and have a great dsay and god bless ::f.b.e.

You could change the receptacle to 4 wire. Or
If you hook a 3 wire cord to dryer, you will need to install the strap that connects to the center terminal(neutral) to the Metal Body of Dryer, Old one may be in place but disconnected when 4 wire cord was installed.
Changing Receptacle may be easier.
4 wire is required for New Construction or Remodeling.