Ah-ha - after way too long, looks like the solution is Server.MapPath
:
Hunspell hunspell = new Hunspell(
System.Web.HttpContext.Current.Server.MapPath("~/Assets/dictionaries/en_us.aff"),
System.Web.HttpContext.Current.Server.MapPath("~/Assets/dictionaries/en_us.dic")
);
To test where the base path is, you can use:
string s = System.AppDomain.CurrentDomain.BaseDirectory;