It should be 4 wire, someone's life could depend upon it.
ie; In my house that was built in 1954 used the copper plumbing as a ground. Worked, but not well.
You couldn't bathe or shower during an electrical storm (Case Iron tub as well). A strike to tree in the yard could follow the plumbing Inside the home.
In the panel box, Plumbing ground and circuit ground went to the grounding bar. One Hot to the breakers.
Standing on the concrete floor in the basement in socks and touching the metal CPU box you could feel a tingle, enough to make untouch real fast. There were NO 3 prong outlets in the house. No chassis grounding what so ever.
That was 110v. Your dryer is 220v. That's enough to stop the heart instantly, cause severe burns, and many other harmful things to the human body.
If it sits on concrete, or ANY chance of water on the floor, make it 4 wire.
Also, check your grounding. Make sure you have double grounding rods in the ground with one continuous copper wire rated for at least 600v.
Not just one rod, and NOT to your plumbing.
Electricity is literately a simple thing once you grab the concept. Deadly if you don't understand it fully.
When I wired my shop it had a wood floor, ran conduit and didn't need to ground the metal boxes, but did. When I extended electricity to the garage with concrete floor it was code to ground the metal boxes. They didn't have code written in 2002 for bare dirt floors, but I grounded the boxes in the storage garage as well. Code has been written since to include grounding of conduit boxes on dirt floors. As you are directly grounded standing on dirt.
The lesson here is if can be grounded in any way, unless you have the proper 4th wire grounding system, lives are at risk.
ie; Dryer is running, kid runs by with a glass of liquid, trips, falls, spills liquid. Goes to stand up and puts hand on metal dryer body while standing in liquid.