the last 2 trips out west from north Alabama we have crossed the river at West Helena, AR, and tried 2 different routes across Mississippi. Both routes add about 40 miles and 1 hour to the path of going thru Memphis.
Coming back from Colorado on 6/22/15 thru Arkansas, we were ready to be home and I decided to just give Memphis a whirl. We hit it about 2 PM on a weekday and had no problems at all. The years-long construction on I-40 between Forest City and Memphis appears to be finished, and other than 4-5 potholes the road wasn't nearly as bad as I-40 thru Oklahoma.
(Note that going east thru Memphis is easier than going west. As you go east, the 2 lanes on the bridge then open up into 4-6 lanes in town. If you are going west, all those 4-6 lanes ultimately have to funnel into 2 across the bridge).
As a lifelong avoider of Memphis, my 2 cents is to give it a try this year....going east.