Since you prefer major highways, I-49 down to I-40, I-40 to Little Rock, I-530 to Pine Bluff, US-65 (mostly 4 lane) to Lake Village, US-82 to cross the Mississippi at Greenville, then across to connect with US-49W at Indianola. Then 49W and US-49 to Hattiesburg, where you can connect with I-59 down to the coast. US-49W and US-49 are also mostly 4-lane divided, with 65 mph speed limits in rural areas.
Making at least 10 trips various routes between NE Oklahoma and Biloxi, the US-65/US-49 route was the best one I found. Some of the cities on the route will slow you down for a while, Greenville and the suburbs south of Jackson being the worst.
There is a short cut around Greenville but not major highways: bear off to right onto SR-454 just after crossing the bridge, right again onto SR-1 coming south out of Greenville, down a short distance to SR-438, east to US-61, south to SR-12 at Holland, then across to pick up 49W at Belzoni. Or once on US-61 you could just follow that down to I-20, take I-20 across to pick up US-49 on the west side of Jackson. Problem with that, I-20 is pretty rough from Vicksburg to Jackson, a closely spaced dips that could cause trailer stability problems; they tossed my car around quite a bit, and it is not very tossable.
Use I-220/I-20 around Jackson on the west side, I-59 to bypass Hattiesburg. Direction you are going, you'll be getting on I-59 anyway.
Tom Test
Itasca Spirit 29B