Since a native method is not forthcoming, I shall post my file based circumvention, for Windows.
You will need to have this utility in the command path (it apparently is stock with Windows 7).
copyUnicode[expr_] := Run["clip <",
Export["$Clipboard.temp", ToString[expr, InputForm],
"Text", CharacterEncoding -> "Unicode"] ];
Usage:
expr = \[Alpha]\[Beta] + Mod[\[Delta]\[CapitalPsi], 2\[InvisibleTimes]\[Rho]^2];
copyUnicode[expr]
This leaves the following text in the Windows Clipboard:
αβ + Mod[δΨ, 2*ρ^2]
Here is a version of the function that holds (does not evaluate) the expression:
SetAttributes[copyUnicode, HoldFirst]
copyUnicode[expr_, form_: InputForm] :=
Run["clip <",
Export["$Clipboard.temp", ToString[Unevaluated@expr, form], "Text",
CharacterEncoding -> "Unicode"]];
Now:
Plot[\[Alpha], {\[Alpha], 0, 10}] // copyUnicode
Puts in the Windows Clipboard:
Plot[α, {α, 0, 10}]