How to plot contours on the faces of a cube

Yet another method:

Let us calculate values of function on appropriate rectangular grids, which we will convert to textures (1 pixel = 1 value). Interpolation between pixels is built-in.

f = 2 #1^2 + 2 #2^2 + #3^2 + #1 #2 &;
PolyhedronData["Cube"] // N // Normal // toTriangles // 
  texturize[f, 50, Hue, Lighting -> "Neutral", Axes -> True]

Here Normal convert GraphicsComplex to separate polygons, toTriangles split polygons to triangles, and texturize put textures on every triangle (f assumed to be Listable)

toTriangles = # /. Polygon[v_ /; Length[v] > 3] :> (Polygon@Append[#, Mean[v]] & /@ 
       Partition[v, 2, 1, 1]) &;

texturize[f_, n_, colf_, opts : OptionsPattern[]] := # /. 
     Polygon[{v1_, v2_, v3_}] :> {EdgeForm[], 
       Texture@ImageData@Colorize[Image@
               f[v3[[1]] + (v1[[1]] - v3[[1]]) #1 + (v2[[1]] - v3[[1]]) #2,
                 v3[[2]] + (v1[[2]] - v3[[2]]) #1 + (v2[[2]] - v3[[2]]) #2,
                 v3[[3]] + (v1[[3]] - v3[[3]]) #1 + (v2[[3]] - v3[[3]]) #2] &
             [#, Transpose[#]] &@
           ConstantArray[Range[-1./n, 1 + 1/n, 1./n], n + 3], 
          ColorFunction -> colf, ColorFunctionScaling -> False], 
       Polygon[{v1, v2, v3}, VertexTextureCoordinates -> {{1 - 1.5/(n+3), 1 - 1.5/(n+3)}, 
     {1.5/(n+3), 1.5/(n+3)}, {1.5/(n+3), 1 - 1.5/(n+3)}}]} /. 
    Graphics3D[data__] :> Graphics3D[data, opts] &;

It is really fast because it uses packed arrays (almost 100 times faster the belisarius's rasterization of DensityPlot)!

Moreover, it is applicable to arbitrary complex mesh:

PolyhedronData["MathematicaPolyhedron"] // N // Normal // toTriangles // 
  texturize[0.7 (#1^2 + #2^2 + #3^2) &, 50, Hue, Lighting -> "Neutral", Boxed -> False]

To obtain "contours" one can use simple discretization with bigger number of points (sometimes it can be faster then ContourPlot):

f = Floor[2 #1^2 + 2 #2^2 + #3^2 + #1 #2, .1] &;
PolyhedronData["Cube"] // N // Normal // toTriangles // 
  texturize[f, 200, Hue, Lighting -> "Neutral", Axes -> True]

The same with the simple cut

Graphics3D[
     GraphicsComplex[
      Tuples[{-0.5, 0.0, 0.5}, 3], {Polygon[{1, 3, 9, 7}], 
       Polygon[{1, 3, 21, 19}], Polygon[{1, 7, 25, 19}], 
       Polygon[{19, 21, 24, 23}], Polygon[{19, 25, 26, 23}], 
       Polygon[{7, 25, 26, 17}], Polygon[{7, 9, 18, 17}], 
       Polygon[{3, 9, 18, 15}], Polygon[{3, 21, 24, 15}], 
       Polygon[{23, 14, 15, 24}], Polygon[{23, 14, 17, 26}], 
       Polygon[{15, 14, 17, 18}]}]] // N // Normal // toTriangles // 
 texturize[f, 100, Hue, Lighting -> "Neutral", Axes -> True]

One can also use MeshFunctions:

Clear[f];
f = {x, y, z} \[Function] x + Sin[5 z] + y^2;
cube = PolyhedronData["Cube", "RegionFunction"];
mesh = 15;
RegionPlot3D[cube[x/2, y/2, z/2],
 {x, -1, 1}, {y, -1, 1}, {z, -1, 1},
 MeshFunctions -> {f}, Mesh -> mesh, 
 MeshShading -> ColorData["Rainbow"] /@ Range[0, 1, 1/(mesh + 1)],
 PlotPoints -> 50, Lighting -> "Neutral"]

---

@ybeltukov discovered that the region expression can simply be True, if the region is the same as the plot region -- neat!:

RegionPlot3D[True,
 {x, -1, 1}, {y, -1, 1}, {z, -1, 1},
 MeshFunctions -> {f}, Mesh -> mesh, 
 MeshShading -> ColorData["Rainbow"] /@ Range[0, 1, 1/(mesh + 1)]]

Another way using textures:

v = {{-1, -1, -1}, {1, -1, -1}, {1, 1, -1}, {-1, 1, -1}, {-1, -1, 1}, {1, -1, 1}, 
     {1, 1, 1}, {-1, 1, 1}};
idx = {{1, 2, 3, 4}, {1, 2, 6, 5}, {2, 3, 7, 6}, {3, 4, 8, 7}, {4, 1, 5, 8}, {5, 6, 7, 8}};
vtc = {{0, 0}, {1, 0}, {1, 1}, {0, 1}};
f[{x_, y_, z_}] := x^2 - y^2 - z^2
q[j_] := MapThread[ Prepend, {{Min@#, Max@#} & /@ Transpose@v[[idx[[j]]]], {x, y, z}}]
ranges[i_] := DeleteCases[q[i], {s_, a_, a_}]
anchoredVars[i_] := Cases[q[i], {s_, a_, a_} :> s -> a]

sides = Table[ Rasterize@  ContourPlot[f[{x, y, z}] /. anchoredVars[i], 
              Evaluate[Sequence @@ ranges[i]], Frame -> False, ColorFunction -> Hue, 
              ColorFunctionScaling -> False,  Method -> {"ShrinkWrap" -> True}], {i, 6}];

Graphics3D[{Black, EdgeForm[None],
           Table[{Texture[sides[[i]]], 
                 GraphicsComplex[v, Polygon[idx[[i]], VertexTextureCoordinates -> vtc]]}, {i, 6}]}, 
           Boxed -> False, Method -> {"RotationControl" -> "Globe"}]

Edit

If you want to use DensityPlot instead of ContourPlot, with PlotPoints->100 you get: