Complete revision: Disclaimer: This is not meant to be a competitor to Tarass' stellar answer. Rather, it is an alternative with lower performance, which however does not require lualatex, which is why I'm posting it. In fact, the only thing it requires in addition to what you've already loaded is the TikZ library calc. This code comes with a new column type which only inserts the TikZ nodes and macro \MeasureLastTable, which you need to place after the table and in which you need to specify the number of columns. This macro determines all relevant length except for one additional vertical shift: the cell contents are not centered vertically and empirically in your example this shift is 2pt. (This number seems to derive from \abovedisplayskip but I did not manage to extract it in a nice way.) After you've "measured" the table, you can cross out single cells with \CrossOut{x} where x specifies the index of the cell in a self-explanatory convention.
\documentclass[letterpaper, 12pt]{article}
\usepackage{array}
\usepackage{tikz}
\usetikzlibrary{calc}
\newcounter{cellindex}
\newcolumntype{C}{>{\stepcounter{cellindex}%
\begin{tikzpicture}[remember picture,baseline=(Cell-\thecellindex-left.base),inner sep=0pt]
\node (Cell-\thecellindex-left){\strut};
\end{tikzpicture}
}c<{\begin{tikzpicture}[remember picture,baseline=(Cell-\thecellindex-right.base),inner sep=0pt]
\node (Cell-\thecellindex-right){\strut};
\end{tikzpicture}}}
\newcommand{\MeasureLastTable}[1]{
\pgfmathtruncatemacro{\LeftCellIndex}{\thecellindex-1}
\pgfmathtruncatemacro{\AboveCellIndex}{\thecellindex-#1}
\begin{tikzpicture}[overlay,remember picture]
\path let \p0 = (Cell-\thecellindex-right.east),
\p1 = (Cell-\thecellindex-left.west),
\p2 = (Cell-\LeftCellIndex-right.east),
\p3 = (Cell-\LeftCellIndex-left.west) in
\pgfextra{\pgfmathsetmacro{\tmp}{(\x0+\x1-\x2-\x3)/4}
\xdef\HorSep{\tmp pt} % horizontal distance between two cell centers
};
\path let \p0 = (Cell-\thecellindex-right.north west),
\p1 = (Cell-\thecellindex-left.south west),
\p2 = (Cell-\AboveCellIndex-right.north west),
\p3 = (Cell-\AboveCellIndex-left.south west) in
\pgfextra{\pgfmathsetmacro{\tmp}{(\y2+\y3-\y0-\y1)/4}
\xdef\VertSep{\tmp pt} % vertical distance between two cell centers
}; % I know that in principle two coordinates would be sufficient here
\end{tikzpicture}%
}
\newcommand{\CrossOut}[2][]{
\tikz[overlay,remember picture]{
\path let \p0 = (Cell-#2-left.west),
\p1 = (Cell-#2-right.east),
\p2 = (Cell-#2-left.north west),
\p3 = (Cell-#2-left.south west) in
\pgfextra{\pgfmathsetmacro{\tmp}{\HorSep-(\x1-\x0)/2}
\xdef\myxoffset{\tmp pt}
\pgfmathsetmacro{\tmp}{\VertSep-(\y2-\y3)/2}
\xdef\myyoffset{\tmp pt}
};
\draw[thick,#1] ([xshift=-\myxoffset,yshift=-\myyoffset+2pt]Cell-#2-left.south west) --
([xshift=\myxoffset,yshift=\myyoffset+2pt]Cell-#2-right.north east);
}}
\begin{document}
\begin{table}[!ht] \centering
\renewcommand{\arraystretch}{1.6}
\begin{tabular}{|C|C|C|}
\hline
AAA & A & AA \\
\hline
AA & AAA & A \\
\hline
A & A & AAA\\
\hline
\end{tabular}
\end{table}
\MeasureLastTable{3} %<- the argument is the number of columns of the last table
\CrossOut{1}
\CrossOut{3}
\CrossOut{5}
\CrossOut{8}
\end{document}
Of course, it will be straightforward to replace the \CrossOut macro by some other fancy annotation.
Original answer: The problem with this is that you can always change the table dimensions, and at a given point your macro will break. In this simple case of a one-column table one can at least make sure that the horizontal positions of the end points of the diagonal lines do not have to adjusted by hand. The vertical positions are still hard coded.
\documentclass[letterpaper, 12pt]{article}
\usepackage{array}
\usepackage{tikz}
\newcommand{\tikznode}[2]{\tikz[remember picture,baseline=(#1.base),inner
sep=0pt]{\node(#1)[inner sep=0pt]{#2};}}
\newcommand{\CrossOut}[2][]{
\tikz[overlay,remember picture]{
\draw[thick,#1] ([yshift=-7pt]#2.south west-|tl.south east) --
([yshift=7pt]#2.north east-|br.north west);
}}
\begin{document}
\begin{table}[!ht] \centering
\renewcommand{\arraystretch}{1.6}
\tikznode{tl}{\strut}%
\begin{tabular}{| c |}
\hline
\tikznode{AA1}{AA1}\\
\hline
\tikznode{A2}{A2}\\
\hline
\tikznode{A3}{A3*}\\
\hline
\end{tabular}\tikznode{br}{\strut}
\end{table}
\CrossOut{AA1}\CrossOut{A2}\CrossOut{A3}
\end{document}
matrixlibrary. In this case, I would be happy to add this to my answer. If you not, I really have no good idea that always works. See e.g. here how messy the guessing of table parameters can get. Needless to say that the proposal there will break if one does something slightly different than what was foreseen by the author of the answer. – Apr 09 '18 at 18:15\struts everywhere? And could you just append such a table to your question? – Apr 09 '18 at 19:41