Kevin,
I am also replacing a four prong cord with a three prong cord.
Can I cut a length of green wire off the old four prong cord and use that as a jumper for from the neutral to the frame?

Theresa,

Please, for future reference do not tag onto an item that is over 3 years old. I'm going to ask that this posting be set up as a new posting. Most responders do not pay attention to items that old.

Are you in your own home or an apartment as was the original poster? I will continue on as if this is your own home

As to your question, Article 300.3 (B) requires that all conductors must be in the same cable. That prohibits you from running a ground wire from the service panel to the dryer receptacle that is outside of the cable sheath.

Now, if the manufacturer of the dryer allows you to, you may be able to make a connection from the dryer neutral to the dryer frame. Only if the manufacturer allows this. That would allow you to use the existing receptacle and you can then transfer the old cord to the new dryer.

They may have designed the device so that such a jumper cannot be installed.
If the manufacturer does not allow this, then if you place the jumper on the frame, you will be putting "Objectionable Current" onto the frame of the dryer.

If that is true, then you must pull a new four wire cable from the service panel to the dryer.