According to zxcvbob :
Minnie Bannister wrote:

But we have subsequently stacked the dryer and the washer, so everything
now has a "real" ground through the washer's power cord, right? (And
doesn't that mean that the neutral and ground are now connected at a
place other than the main panel?)

No. Your dryer has 2 hots and a ground and no neutral. An exception in
the old code allowed you to use the ground for the unbalanced portion of
the dryer load (for the timer, buzzer, light, and maybe the motor).

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

Your
washer has a hot, ground, and neutral, and its neutral is isolated from the
frame.

Would there be any advantage in replacing the 3-pin dryer outlet by a
4-pin and replacing the 4-pin cord?

I would leave it alone. I assume you know you cannot just replace the
outlet unless you run a big (probably #10 copper) neutral wire? This means
running a whole new cable to the dryer, unless the wires are in conduit.
If you're willing to rewire it properly, it would be marginally safer to
have 4 wires. But not much safer as long as the neutral connections are
good and tight in the 3-wire hookup.

Here's something that may put this into context:

If I understand the NEC rules maze correctly, if you move into a house that
has a three wire stove connection, at most, NEC requires you to put a
receptacle on it. You do not have to upgrade to four wire. Which is why
both 3-wire and 4-wire receptacles and stove/dryer plugsets are available,
and why stoves and dryers are convertible.

In contrast, by CEC rules, if you move into a house that has a three wire
stove connection, not only do you have to install a receptacle, you must
upgrade to 4 wire. Which is why three-wire stove/dryer plugsets are _not_
available here (and three wire receptacles only available for non-dryer/stove
applications - pure 240V only - you can't feed a neutral on a 3 wire 240V
circuit).

[For compability with US standards purposes, most of our stoves and dryers
are convertible too, but it's specifically not listed in the instructions,
or there are disclaimers it doesn't apply to Canada or covered by the
generic "consult your local code!" fine print.]

The NEC doesn't think the risk is high enough to require 4-wire upgrades to
3-wire systems. The CEC does.

The NEC doesn't think the risk is high enough to require mandatory refit
on existing installations (except for new work). The CEC agrees here.

Whether you want to go to the upgrade-to-4-wire CEC rules is something you
have to answer for yourself.

I would think there's a _slightly_ increased risk with stacking a ground-neutral
bonded appliance with a ground-neutral separate appliance. But minimal at worst.
--
Chris Lewis, Una confibula non set est
It's not just anyone who gets a Starship Cruiser class named after them.