Updating 2 prong 3 prong outlets
You could replace the cord on the dryer, but you'd have to bond the chassis of the dryer to make that acceptable to current NEC codes (Article 250.140).
However, this will require you to run new cable to the receptacle, since you'll need a cable with a ground.
Be sure, though, that there aren't indirect ground plug connections, such as via the sheath on BX cable. The NEC requires that three prong receptacles without ground that are protected by GFCI must be labelled as such.
This schematic for an electric dryer might help you understand how the dryer could be wired (depending on make/model/manufacture date).No matter what your specialty is you'll find that Electrician is a great community to join. Always use proper safety precaution and reference reliable outside sources before attempting any construction or remodeling task!Welcome to the DIY Forums where you can ask questions or find answers on anything related to "Do it yourself" or home improvement.(FWIW, a new cord seems to run around - (as of 2015), depending on how long you need it and where you buy it.) Whirlpool has detailed wiring instructions so you can make sure you are wiring it properly.Other manufacturers will likely have similar instructions. Contrary to people here, they recommend not updating the outlet to NEMA 14 yourself, but having a licensed electrician do it if that's the way you want to go.