Unless things have changed recently, this is not necessarily true. The river bottoms of many, if not most rivers in Texas, particularly if the mouth of the river is over 30 feet wide, is public property. Exceptions may be land originally deeded from Spain and Mexico that can privatize the river bottoms also.
Hunting public river bottoms below the cut bank, is perfectly legal in many cases, again unless things have changed recently. To be sure you should check with your local game warden.
For more on this read...
http://www.tpwd.state.tx.us/texaswater/rivers/
riddellindex.phtml