Stick with hardwoods, and avoid any woods known for oils (i.e. teak) or intense characteristics (i.e. aromatic cedar or mahogany). My rule of thumb is to stick with woods known for use in foods or brewing. Birches and maples are all very neutral woods used for things like toothpicks, cutting boards, etc. Various oaks are obviously associated with barrel-aging properties. Beeches, applewood, cherrywood, and hickory are all associated with smoked malts and other smoked/roasted foods.
My mash paddle (sorry, no pic) is carved of a 3/4" thick cherrywood plank with hop pictures carved in each face. No finish on it - I'd rather not add the foreign substances.