Evaluate calculations on label

I would evaluate it as part of the loop. For example,

\documentclass{standalone}
\usepackage{tikz}
\begin{document}
\begin{tikzpicture}[y=-1cm]{every node/.style={font=\scriptsize}}
  \draw [step=.1cm,very thin,black!10] (0,0) grid (6,6);
  \draw [step=2cm,thin,black!50] (0,0) grid (6,6);
  \draw [very thick,black] (0,0) rectangle (6,6);
  \draw [->,very thick] (0,0) --  (6.5,0) node[below] {$X$};
  \draw [->,very thick] (0,0) --  (0,6.5) node[right] {$Z$};

  \foreach \x in {0,1,...,2} {
    \foreach \y [evaluate=\y as \z using {int(\x+3*\y)}] in {0,1,...,2} {
      \node (node) [label={[yshift=-.6cm, xshift=.3cm]:\z}] at (2*\x,2*\y) {};
    }
  }
\end{tikzpicture}
\end{document}

Alternatively, you could calculate \z within the loop. For example,

\pgfmathsetmacro\z{int(\x+3*\y)}

before typesetting the \node would effectively do the same thing.

EDIT [Response to follow-up]

You could just put the labels inside the nodes and set the anchor as you wish, rather than messing around with shifts. For example,

\node  [anchor=north west] at (2*\x,2*\y) {\z};

produces

You can evaluate expressions with the suitable keys in a for loop. Hence if you change your code to

    \foreach \y[evaluate={\yi=int(\x + 3*\y);}] in {0,1,...,2} {
        \node (node) [label={[yshift=-.6cm, xshift=.3cm]:\yi}] at (2*\x,2*\y) {};

Here int is needed to return integers instead of 1.0, 2.0... etc.