How can I customise memoir's index to have more than two columns?

Use the idxlayout package. It allows for indexes with more than two columns and is responsive to the index-related options and commands of the memoir class.

Disclosure: I'm the package author.

\documentclass{memoir}

% \usepackage{makeidx}% Emulated by memoir
\makeindex

\usepackage[columns=3]{idxlayout}

\begin{document}

\chapter{foo}

Some text.\index{Alpha}\index{Bravo}\index{Charlie}\index{Delta}\index{Echo}\index{Foxtrott}

\printindex

\end{document}

^{Sorry for ressurrecting an old thread, but I'd like to provide another suggestion for completeness' sake. :)}

The most recent version of the imakeidx package is memoir-compliant. We can easily use it to customize our index, including the number of columns. :)

\documentclass{memoir}

\usepackage{imakeidx}[2012/05/09]
\makeindex[columns=3]

\begin{document}

\chapter{Hello}

Some text.\index{Apple}\index{Apricot}\index{Avocado}\index{Banana}
\index{Bilberry}\index{Blackberry}\index{Blackcurrant}\index{Blueberry}
\index{Currant}\index{Cherry}\index{Cherimoya}\index{Clementine}
\index{Date}\index{Damson}\index{Dragonfruit}\index{Durian}
\index{Eggplant}\index{Elderberry}\index{Feijoa}\index{Gooseberry}
\index{Grape}\index{Grapefruit}\index{Guava}\index{Huckleberry}
\index{Jackfruit}\index{Jambul}\index{Kiwi fruit}\index{Kumquat}
\index{Legume}\index{Lemon}\index{Lime}\index{Lychee}\index{Mandarine}
\index{Mango}\index{Melon}\index{Nectarine}\index{Orange}\index{Peach}
\index{Pear}\index{Pitaya}\index{Physalis}\index{Plum}\index{Pineapple}
\index{Pomegranate}\index{Purple Mangosteen}\index{Raisin}\index{Raspberry}
\index{Rambutan}\index{Redcurrant}\index{Salal berry}\index{Satsuma}
\index{Star fruit}\index{Strawberry}\index{Tangerine}\index{Tomato}
\index{Ugli fruit}\index{Watermelon}\index{Ziziphus mauritiana}

\printindex

\end{document}

The output: