It turns out it is just a case of passing in the date-time string as usual (e.g. 2013-09-23T16:47:00) in the URL, but you need to add requestPathInvalidCharacters to httpRuntime in web.config, i.e.:
<httpRuntime requestPathInvalidCharacters="">
so that the colons in the date-time string don't cause the A potentially dangerous Request.Path value was detected from the client (:).
error to occur.
This is a bit unsafe though, so alternatively you can pass your parameter through in the query string and they will model bind correctly that way.