I can see a problem if you have a left turn lane and a left + straight ahead lane. I would then get to the right of the left turn lane, but when the traffic allows me to turn I would first go straight ahead before starting the turn so I am in the RH turn lane as I am making the turn.
use a mirror to look for gaps in the traffic. Traffic lights on the 6 lane street should create gaps across all 3 lanes in your direction. This will make it easy for you to get to the middle of the road. If you have to do this early, motorists wont have a problem passing you on your right.