The instructions on the dryer should tell you exactly what is needed. Do you have a 4-prong outlet for the cable? If so, then that ground might be needed internally. If you are trying to install a new washer (or stove) and only have a 3-wire plug, they'll allow you to do it with a 3-wire cord. When sold, the dryer or stove has the neutral and ground separated in the thing, each going to a prong of the 4-wire plug. If you only have a 3-wire plug, the ground and neutral inside need to be connected together. So, if you have a 4-wire plug, it is already grounded properly.
Attaching the ground lead to something that is grounded, maybe the screw of the receptacle for the washer, may be acceptable. It is an extra safety, and (I think) you should be safe without it, but it doesn't hurt.