Yes, that's the problem with hard wood. It is very heavy. If you could section those thicker pieces so that they are only 1-2" thick, that would give you the "wood" on one side and the bark on the other. The flat wood-side could then be hung or just placed on a surface, allowing the plant to grow more upright.
Here are some examples of various woods, all sectioned (except the cork) to show the character of the wood on one side and leaving the bark side untouched. I generally mount my plants on the bark side but sometimes will mount them on the finished side just to show off the wood.
cypress, oak, cedar, cork, cypress knee, heart-pine
white oak
red oak
cypress
oak (2)with cypress in the middle
cypress, oak, cedar, cork, cypress knee, heart-pine
white oak
red oak
cypress
oak (2)with cypress in the middle
