According to zxcvbob :

Most dryers have 120V motors and timers, so they "need" neutrals.

They need a neutral, but a 3-wire cord doesn't give them one. It dumps
the unbalanced return voltage on the ground. A small fixed load, and a
very large low-resistance ground. [I didn't design it] Should be OK
with copper wires and tight connections; I wouldn't trust it with
aluminum wire even with proper terminations. I would use aluminum for a
4-conductor dryer or range circuit.

[now that I shot my mouth off, I gotta go doublecheck whether that
grounded wire is technically a ground or a neutral]

This is really more of a terminology quibble than anything else, for
what really matters is what the third wire is being used _for_. The
white wire in your house isn't technically a neutral _either_. That's just
the term everyone uses. It's official name is "grounded conductor".

It's used as _both_ in the case of a 3-wire stove or dryer. It's
being used as frame ground _AND_ neutral return. This starts to become
obvious when you start looking into (older) code restrictions where, for example,
you (normally) couldn't 3-wire a stove or dryer from a subpanel that has split
neutral and ground.

My dryer has an old 3-wire outlet; it's connected directly to the
service panel with a short length of rigid metal conduit, so even if the
grounded wire were to somehow come loose I'd have a good equipment
ground through the conduit so I haven't bothered to change it to 4-wires.

This is slightly better than unconduited 3-wire, in that you
have better connections to the ground. However, if the neutral
separates in your panel or on the pole, the frame of your stove _still_
goes hot. The problem is that the frame is connected to the device neutral,
not how well it's connected to the system ground.

Because the fact of the matter is, if your neutral separates in the panel,
AND if you have a neutral-ground interconnect anywhere in your house,
_every_ grounded object in your house can potentially go to damn close to
line voltage. By NEC rules, ground electrode conductivity is not necessarily
high enough even to trip a 15A breaker, let alone the mains.
--
Chris Lewis, Una confibula non set est
It's not just anyone who gets a Starship Cruiser class named after them.