This is a graphics software question; please answer only if you are
experienced with such software. I use Adobe Illustrator, which
apparently cannot do what I need.

The problem: Picture a canvas with several hundred circles in specific
locations, each circle defined as an object. I want to separately
design a different object, let?s call it a triangle, and replace each
circle with the triangle with a single command. The result would be a
canvas with several hundred triangles in the locations formerly
occupied by the circles.

Question: What graphics program(s) that can substitute one object for
another on a global basis, as explained above.

Actually, Illustrator can do this with symbols.
In the situation you described you could define your circle object as
a symbol using the symbol palette (Window>Symbols).  You could then
duplicate the symbol as many times as you like and edit the original
symbol (to a triangle) and all of the circles would be replaced. I
believe symbols were introduced as a new feature in Illustrator 10, so
you'll need version 10, CS, or CS2 to use them.

The help menu has more detailed information on working with symbols
(Help>Illustrator Help).