Yes, I know the concept of perpetual motion violates the known laws of
physics. Yet, the movement of the planets around the sun is a sort of perpetual
motion, agreed that nothing is forever. Could it be that gravity is the key to
perpetual motion? Is there any serious thought being given to the possibility
of constructing a perpetual motion machine?

Well what you're thinking is that since the planets keep moving, they
must have some force acted upon them which requires energy. This is
not true. The planets move because there is not force stopping them.
(Newton's Law of Inertia)

I think Monroe's questions stated that the planets are perpetually
moving without force upon them pushing them in the direction of their
rotation around the sun and that is why perpetual motion might exist. 
Piotrr, there are atleast 2 forces at work with the planets
1) gravity from the sun
2) centrifugal force from their rotation around the sun

I'm certain their are others (a moon's gravity pulling on the planet
or other distant stars pulling slightly on them... which has more
effect on outer planets like Pluto of course), but the 2 listed are by
far the greatest.

Monroe, to answer your question, gravity can definately be harnessed
to some degree to help a motion machine perform more efficiently,
however...  if anything is moving on or around the Earth then it has
friction either on the ground and/or in the air that it must overcome
and that requires force.  There are of course natural ways to harness
that force needed like solar power or wind power or perhaps in the
near future hydrogen power which are all seemingly endless resources
we have.

Entropy

Your answer is in the first two answers, and the solution is in the
last! there are in fact zero net forces in action in the movement of
the planets, gravity is not really a force just things falling
unavoidably through space-time to increase the mass of the object
towards which they are falling.  Centrifugal force only exists in the
minds of college students whilst the centripetal force of the planets
is merely a product of inertia caused by the planets falling
unavoidably through space time.  There is no serious research into the
subject because of entropy and the laws of thermodynamics.

somerset13: Before I asked the question, I had a grasp of most of what
was said in the comments. Just wondering if there was some offbeat
thinking which might posit some weird physics allowing perpetual
motion. After all, some theories in astrophysics are so mind-boggling
as to be completely incomprehensible to ordinary mortals. Is perpetual
motion less strange than parallel universes?
monroe22

monroe: Perpetual motion is something hated by all college lecturers,
not because of what it could be! but because of all the questions
asked about it by students and the fact that they had it beat out of
them at college.  The problem with perpetual motion is that if it
could or did exist it would be useless.  It's very nature infers that
any energy created must go back into the system to keep it running,
allowing for no losses whatsoever!!  An inquisitive mind should be
investigating ways in which we can harness energy, putting the minimum
in to get the maximum out is the key.  Imagination and science will no
doubt provide the answer.  You are right however, the more
controversial and often most modern theories seem bizarre and I often
wonder if scientists create new ones without fully researching or
understanding the old ones, ahh! if only Einstein had lived for
another hundred years.

PS: Monroe since I know not your age/ engineering knowledge/ interests
etc this may be inappropriate for the question, but if the subject of
energy production interests you look into sterling engines in general
and this link talks of an acoustic type sterling engine.  These
engines have been around for a long time and are incredibly efficient.
www.electrifyingtimes.com/eng.html

somerset13: I am 76, a retired chemist, some knowledge of chemical engineering.
Thanks for the link.
monroe22

Some sub-atomic bits move in perpetuity but unceasing motion is not
what is intellectually implied by the term. It has to be a system from
which energy can be extracted. The distinction is lost to most of
course.

You can?t get owt for nowt, so conventionally, perpetual motion is
impossible. Even gravitational systems slow down the planet?s motion.
But, theories on vacuum energy indicate a ?maybe? whereby virtual
particles zap in and out of existence. Blame Heisenberg. The temporary
violation of mass-energy conservation might be utilised to build the
?impossible machine? but what one would do with the negative energy
left over I don?t know.

Best

To all: OK, I give up. I knew in advance, if anyone commented, the
reply would be: Impossible. I was hoping, in my dotage, that some
unsung genius would say, maybe, or even, yes. It would be beautiful to
have a non-polluting, infinitely renewable energy source. Hydrogen? A
definite maybe.
monroe22

It would indeed be nice to have an 'infinititely renewable' source of clean energy.

However, I would be more than happy to settle for an 'ample supply'
eg: I would be just as happy with $1Bn as with a Googol of $

We are literally swimming in energy, there is the Sun, tidal movement
due to the Moon, electro magnetic fields everywhere, not to mention a
rather hot core of the Earth. (note: I have excluded wind farms, which
I consider a rather silly idea).

Hydrogen is not really a source of energy, unless one is going to
crunch it or combine it (fision or fusion) both of which activities
are rather dangerous, and the latter is (I think) largely theoretical.

It is more a 'store of energy' or a means of carrying it around.
Rather like a can of beans on a high shelf has considerable potential
energy, as can be demonstrated by knocking it off and letting it land
on ones foot.

Personally I reckon that we should be looking into tidal energy, as it
is proven technology - Canada supplies North N. America with huge
quantities of electricity from hydroelectric dams. Clear water, salt
water, who cares.

Realistically, while we have vast reserves of 'concentrated' energy in
oil, there is little (direct) incentive to look for alternatives.

http://popularmechanics.com/automotive/auto_technology/2000/12/hydrogen_fuel_cell_car/print.phtml

Hydrogen cars are well on their way.  Honda seems to be the
alternative fuel leader in todays auto market and they're hoping to
have hydrogen cars for sale by 2010.  I'd imagine that the
infrastructure will be in place for hydrogen to be commonly used by
2015, and from there I don't see why hydrogen won't surpass gas as the
leading source for auto fuel by 2020.

How about a hydrogen power plant supplying 10 times what a coal plant
produces but with almost 0 pollution?  I do believe hydrogen will make
huge impacts on the US economy and lifestyle within another 20 years.

THE BIGGEST BREAKTHROUGH SINCE SLICED BREAD   OH BABY!

jack_of_a_few_trades: Sure, hydrogen looks nearly perfect, on paper.
The huge problem is that it must be extracted from something. That
requires energy, and since no mechanical process is 100% efficient,
there will be a net energy loss.
Electrolysis of water produces hydrogen and a lovely by-product,
oxygen, if you have no objection to at least 1000 (yes, 1000) nuclear
power plants needed to supply enough hydrogen to power US autos. Of
course, we could build 1000 conventional power plants using fossil
fuel...oops, strike that. The two most used sources for hydrogen today
are...guess what...petroleum derivatives.
  Fuel cells for autos? Maybe after much more research. Today, it
takes about 30 minutes for fuel cells to warm up and power a $100,000
car. Bicycles make more sense. Transporting liquid hydrogen to
distribution centers should be great fun when a tanker crashes,
producing the equivalent of huge bomb.
Once you get past the chatter of the enthusiasts, it becomes clear
that hydrogen power contributes more problems than it solves.
monroe22

You have heard a number of "scientific" answers, now here is the
"engineering" answer.  Although there are valid physics laws that say
that Perpetual motion is not possible, if you take a more pratical
perspective, perhaps it IS possible...sort of.  Let's redefine
Perpetual motion not as more energy out than in but rather as useful
energy out with no cost for any input.  In that regard, ocean waves
might be seen as Perpetual motion for all pratical purposes and a
machine that harnessed ocean waves might be seen as Perpetual motion. 
Here are several others:

The heat differential between two points can use convective flow to
move energy - everyone knows about geothermal but waht about from
ocean bottom to surface?  High up in the atmosphere to a lower level?

Air flow can also be seen as Perpetual motion for all pratical
purposes and I don't mean wind energy.  If a pipe runs from the bottom
of a mountain to the top, the atmospheric pressure difference will
create an airflow that will persist as long as the mountain is there. 
Now isn't that Perpetual motion for all pratical purposes?

I am not sure about the pipe on the side of the mountain, the weight
of the air inside the pipe would be the same as the air outside.

Dropping warm top water to the cold ocean bed sounds interesting.

As Monroe22 points out, hydrogen is a lousy way of storing energy, it
is volatile and needs to be kept under pressure. A long time ago I
read something about an 'earth' that would absorb hydrogen under
atmospheric pressure, but that seems to have gone quiet.

While burning hydrogen to produce water sounds a clean way of
obtaining power, air is 80% nitrogen, so it might not be quite so
clean in practise. Realistically we would probably be better off using
ethanol or methanol in internal combustion engines. I'm rather unsure
about the CO2 scare.

My understanding is that the major problem is 'storing' energy, so
that it can be tapped at the right time, I've vague memories of water
being pumped uphill into dams so that it can be let back down again
when demand peaks.

Water pumping for energy storage alive and well in Scotland and Wales.

The holy grail of fusion energy is still beyond present technology,
but exceedingly inefficient fusion is actually very cheap and easy.
Search string :- farnsworth "nuclear fusion"

Incidentally, chances are that a crt will very occasionally produce a fusion event.

Best

Thanks Guzzi,
Two interesting points.