How do I split a views page into multiple columns?

You need to use a theme function to override the default output. If you're just using a teaser view for this, then you want to use (assuming phptemplate)

function phptemplate views_view_teasers_VIEWNAME

The function itself is fairly straightforward, it (eventually) just iterates over the $nodes array, calling node_load and then node_view.

What you'd need to do is set up some divs and some CSS to lay it out 3 across.

Try the views bonus pack + panels.module