You can clean a wooden pipe much like a glass pipe. Just make sure you will rinse it more as the wood can soak up some alcohol, and more importantly let it fully dry before use.
Alcohol and a pipe cleaner work great. You really do not want to get alcohol on the outside of the pipe though. It could dry out the wood and make it look bad. If you do get alcohol on the outside, it is a good idea to re-oil the wood to keep it lasting longer and from drying out the wood too much.