With or without bark, wood will burn less efficiently when it is cold, which creates that thick smoke. Once the wood get's up to temperature, it will burn more efficiently -- which means it will burn cleaner.
The best way to avoid that smoke is to add small pieces of wood to your hot coals. You can also help the process by pre-heating your wood in the sun, or on top of your cooker. Use smaller pieces of wood, which means you will have to add wood more frequently, but your lower wood to coals ratio will give you a cleaner smoke.