How would food be refrigerated in a dining car kitchen in one of the
long-distance luxury passenger trains in the 1930s and 40s (say, Santa
Fe's Super Chief on the Chicago-LA route)? Did they use ordinary ice,
dry ice, or even mechanical refrigeration? When did changes in cooling
technique occur? References?

Mechanical refrigeration is older than you might think. It was in
commercial use in the 19th century.

"James Harrison, a Scottish journalist, moved to Australia from
Glasgow and set about designing his own refrigeration machine.  In
1855 he succeeded in creating and patenting an ether liquid-vapour
compression fridge.  The compressor worked by exerting pressure on a
refrigerant gas, forcing it to pass through a condenser, where it
cooled down and liquefied. The liquid then circulated through the
refrigeration coils and vaporized again, cooling down the surrounding
air.

Australia was in desperate need of refrigeration because of its lack
of natural ice needed for keeping food cool.  Harrison was convinced
that the economic salvation of Australia lay in the marketing of her
millions of sheep and cows to the millions of Europeans.  But without
refrigeration, it was impossible to ship the mutton across the 100
Degree-plus Equator.

Harrison spent his last penny to equip the Norfolk ship with a
chemical freezing mixture for twenty tons of beef and mutton at
Melbourne.  But when the meat arrived in London, it was discovered
that the chemical tanks had leaked and ruined the entire cargo.  As a
result, James Harrison went into bankruptcy, even being forced to sell
his successful newspaper business.  Though Harrison was financially
devastated, he did open the door for the economic salvation of
Australia.  Other successful refrigerated voyages followed, which
finally convinced Europe that Australia had something to offer, and
that frozen food could be both safe and delectable."

http://www3.bc.sympatico.ca/st_simons/cr0003.htm

By 1950, mechanical refrigeration was certainly in use:
http://www.wataugavalleynrhs.org/index.php?template=moultrie

This site about roadside "diners" from the 1920s mentions that an
electric refrigerator could be an extra, suggesting that at that time
icebox refrigeration was still an acceptable alternative.  But note,
these diners were never derailed RR dining cars.
http://www.nydiners.com/mulholland.html

Back in Australia, this site describes a 1929 RR car as having
refrigeration, but since it also mentioned that a refrigerator was
installed in the kitchen in the 1960s, the distinction is unclear. 
Maybe the later was a gas-powered one.
It could be thought that in that climate and in the remote area,
access to ice would have been rare, so that the original fridge was
electric or mechanical, but the question is open.
http://www.prr.org.au/cms/content/view/141/89/

But the above may have been referring to air conditioning, since this
site describes such from the 1920s:
http://www.atsfry.com/Passenger/steameject/steamejector.htm

But if they could do that, they could have refrigerated food the same way.  ??