This is based on Rahul's ideas, but a different implementation:

contourRegionPlot3D[region_, {x_, x0_, x1_}, {y_, y0_, y1_}, {z_, z0_, z1_}, opts : OptionsPattern[]] := Module[{reg, preds}, reg = LogicalExpand[region && x0 <= x <= x1 && y0 <= y <= y1 && z0 <= z <= z1]; preds = Union@Cases[reg, _Greater | _GreaterEqual | _Less | _LessEqual, -1]; Show @ Table[ContourPlot3D[ Evaluate[Equal @@ p], {x, x0, x1}, {y, y0, y1}, {z, z0, z1}, RegionFunction -> Function @@ {{x, y, z}, Refine[reg, p] && Refine[! reg, ! p]}, RegionBoundaryStyle -> None, opts], {p, preds}]] 

(update - added RegionBoundaryStyle -> None which is required in v12.2)

Examples:

contourRegionPlot3D[ (x < 0 || y > 0) && 0.5 <= x^2 + y^2 + z^2 <= 0.99, {x, -1, 1}, {y, -1, 1}, {z, -1, 1}, Mesh -> None] 

contourRegionPlot3D[ x^2 + y^2 + z^2 <= 2 && x^2 + y^2 <= (z - 1)^2 && Abs@x >= 1/4, {x, -1, 1}, {y, -1, 1}, {z, -1, 1}, Mesh -> None] 

contourRegionPlot3D[ x^2 + y^2 + z^2 <= 0.4 || 0.01 <= x^2 + y^2 <= 0.05, {x, -1, 1}, {y, -1, 1}, {z, -1, 1}, Mesh -> None, PlotPoints -> 50] 

How it works

Firstly LogicalExpand is used to split up multiple inequalities into a combination of single inequalities, for example to convert 0 < x < 1 into 0 < x && x < 1.

Like Rahul's code, an inequality like x < 1 is converted to the equality x == 1 to define a part of the surface enclosing the region. We do not generally want the entire x == 1 plane though, only that part for which the true/false value of the region function is determined solely by the true/false value of x < 1.

This is done by plotting the surface with a RegionFunction like this:

Refine[reg, p] && Refine[! reg, ! p] 

which is equivalant to the predicate "reg is true when p is true, and reg is false when p is false"