For xelatex and lualatex the default font family is Latin Modern. The font does not cover a wide range of Unicode. Why not extend the font characters to cover other languages' script? or define a default font family with a greater coverage of Unicode like DejaVu font family? Thus one can add some non-Latin characters directly.
\documentclass[12pt]{article}
\textheight=2cm
\begin{document}
\makeatletter
\edef\textFontName{\fontname\csname
\f@encoding/\f@family/\f@series/\f@shape/\f@size\endcsname}
\makeatother
See\footnote{Text Font: \texttt{\textFontName}}
μ % <- lmodern font do not contain this
\end{document}
