It should be easy enough to determine which direction the fans are blowing
by spraying a mist of water and seeing which way the mist blows.  The fans
should be blowing into the engine compartment and the fan shrouds should be
in place.

Some other things to check:

The coolant should be translucent and be a 50-50 mix of anti-freeze and
water.  If it is not translucent or has a muddy appearance, the cooling
system should be flushed and the coolant changed.  If there is oil in the
coolant, or if the engine oil looks like a milkshake, the head gasket may be
leaking or the head may be cracked.

The radiator should be clean inside, and the outside should be free of
debris.  If the car has air conditioning, the outside of the condenser
should also be free of debris.

When the cooling system was filled, the heater should have been set to full
hot so that no air is trapped in the system as coolant is added, and the
overflow bottle should also be filled.

The upper and lower radiator hoses should be pretty solid and not collapse
when coolant is flowing.

When the thermostat was installed, the spring should be facing down.

Other things that can make a vehicle overheat are a restriction in the
exhaust, inoperative EGR system, over-advanced ignition timing, slipping
timing belt (not too likely), and incorrect spark plugs.

 Besides the stuff Ray brought up...

 Make sure they used a Toyota Factory thermostat, and it's mounted
the right way.  

 We hear wierd things about aftermarket thermostats - contrary to
common sense, if you let the coolant flow too fast through the heads
and cylinder jackets it can't pick up the heat properly.  And the
accelerated flow affects where the coolant swirls and eddies, and the
flow doesn't go where the engineers intended it to.

 The next thing before you get too involved tearing the whole car
apart is to make sure that the dashboard temperature gauge isn't lying
to you - it happens.  It could be a bad gauge movement or engine
mounted temperature sender, or a bad ground from the sender body
threads to the intake manifold to the engine block.

 You do NOT want to put 50 layers of Teflon Tape on the gauge sender
or fan thermostat threads - The single-lead senders and switches count
on making some metal-to-metal contact for the ground.  This is where
you use Rectorseal #5 pipe dope or the old black Permatex sealant
instead of Teflon Tape.  Same thing with gooping on Silicone Sealant
and accidentally electrically insulating the water outlet adapter from
the engine block.

 You can get an aftermarket coolant temperature gauge and lash it up
"temporarily" to see what is really going on.

 You could have one or both electric cooling fan motors starting to
go bad - they have bronze oilite bushings on the motor shafts, and
they do wear out over time.  And a symptom would be they run fine for
a while, but when they get hot they'll start moaning or whining, the
motor shaft heats up and they start seizing in the bore.

 Check the ampere draw of each fan motor against the ratings, a bad
motor can draw too little power (failing windings) or too much.

 Also, get the radiator checked - with an Aluminum core and plastic
crimped-on tanks they can't easily disassemble it like the old Copper
jobbies, they can only chemically clean it out and do a flow test on
the bench.  If it fails the visual check or flow test, replace it.

 The aluminum ones are cheap enough to be considered 'disposable',
but if you will keep the car a while try to find an old-style Copper
radiator.  They can be unsoldered and rodded out every few years.

 Remember the old Overheating Emergency tricks:

 Trick One:  Roll your windows down and turn the heater up all the
way, full hot, full fan.  That's adding some additional radiator
surface to the system, but it dumps the heat in the car instead of
under the hood.

 Trick Two:  Carry a 2-gallon garden sprayer, the pump-up kind,
filled with clean water.  If the engine is overheating, set the spray
nozzle wide open and spray a heavy mist of water into the radiator.
The evaporative cooling should drop the temperature like a rock.

 And if the temperature still doesn't come down, you have other more
complex problems.  Collapsing or blocked radiator hoses (they can
develop internal "aneurysms" of the liner layer that blocks the flow),
corroded water pump impeller that is not moving the coolant, the water
pump was rebuilt with a reverse rotation impeller from another car
model...

 In other words, never rule out stupidity at any stage.  The parts
man can hand you the wrong item even though it fits in the hole.

    --<< Bruce >>--

Use a thermometer to check the coolant temp as the engine warms up.  The
coolant fan should come on at 199F and turn off at 181F.  See if the
actual temp corresponds with the temperature gage position.

I am the original poster of this thread. I wanted to let everyone know
the outcome of my "adventure". To make a long story short, I took the
car back to the Toyota dealer who did the head gasket work. After three
days of testing they gave me the car back saying they could not find
anything wrong. Everything in the cooling system was operating as it
should. I drove the car for less than a week and it started to overheat
again. I happened to be on my way to the dealer for state inspection so
the mechanic was able to see the car at the actual point of
overheating. Still they couldn't find anything wrong! Finally the
service technician suggested, as a last remedy, a power flush of the
cooling system. The dealer didn't provide that service but recommended
a nearby radiator shop. Eureka! The radiator shop owner said the
radiator (which was only about a month old) was full of gunk! He was
able to successfully flush out the radiator and the engine. I have been
running for a week now and the temperature gauge needle sits
comfortably just below the halfway point. I learned a lesson from this
episode. If you are going to have a radiator replaced it's a good idea
to have the entire cooling system flushed. Think of it as cooling
system insurance; especially when buying a car from a private seller
where the service history is unknown.

Mike