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What's the typical way to make Mathematica plots use purely cyan and black? Maybe there's a fairly standard method, since publishers often require this?

It also seems like some plot types take PlotStyle argument while others (for example, ContourPlot3D) take ColorFunction, so maybe the method differs depending on the plot type?

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    $\begingroup$ You can use a color function like Blend[{Black, Cyan}, t] for plots. For images, you can use that with Colorize[]. $\endgroup$ – J. M. will be back soon♦ Jun 15 '15 at 22:28
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The solution is to use

Blend[{Black,Cyan}, <appropriate fraction>]

Two examples:

DensityPlot[
 Sin[x] Sin[y],
 {x, -4, 4}, {y, -3, 3},
 ColorFunction -> (Blend[{Black, Cyan}, #] &)
 ]

Mathematica graphics

Plot[
 {Sin[x], Sin[2 x], Sin[3 x]},
 {x, 0, 2 Pi},
 PlotLegends -> "Expressions",
 PlotStyle -> {
   Blend[{Black, Cyan}, 0],
   Blend[{Black, Cyan}, 0.5],
   Blend[{Black, Cyan}, 1]
   }
 ]

Mathematica graphics

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