Mountain roads in western Arkansas tend to be more about tight and twist roads, using switchbacks rather than steep grades to get across the mountains. Where grades are steep, they also tend to be short.
You can get through Arkansas to Missouri without traveling any mountain roads, depending on where you enter the state, where you want to leave it, and what you want to visit while there. I do it often, Fort Smith to Greenville in Mississippi, following the Arkansas River valley into Little Rock, then across the Missippi Delta (which makes about a third of the state flat).
Direction you are going, you could take I-30 to Little Rock, I-40 up to West Memphis, I-55 to Normal, I-39 up to Wisconsin (or if going to Lake Michigan, I-57 out of Sikeston, Missouri avoids every big city between Memphis and Chicago). But with this route, by avoiding the hilly parts of Arkansas and Missouri means also avoiding most of the interesting destinations in both states.
Actually, because I hate the traffic on I-40 and the mess around Memphis, I would personally take US-67 NE out of Little Rock, connecting with US-60 in southern Missouri to get back to the junction of the Interstate highways at Sikeston. I use US-60 across Missouri quite often, to bypass St Louis, when heading toward Kentucky or southern Illinois. I have other routes through Missouri to bypass St Louis on the north, when headed toward Iowa, Wisconsin, Chicago, Northern Indiana or Western Michigan. These don't go through Arkansas because I am starting from NE Oklahoma.
The mountains you are likely worried about could be the Ouachita, which spread from Hot Springs to McAlester, and can be crossed North-South on US-259 through Oklahoma or US-59 and US-71 in Western Arkansas. Those two roads get interesting and can be slow traveling.
North of I-40 and the Arkansas River, you get into the Ozarks, which are more gentle, and there are two good North-South routes: I-540/US-71 out of Fort Smith, on the west side of the Ozarks, and US-65 out of Little Rock, on the east side of the Ozarks.
If you want to go through the Ozarks rather than around them, then it is SR-7 through Russellville. As with the stretch from Hot Springs to Russellville, it is more twisty than it is steep, and the country is beautiful. Missouri has no mountains, really. Their Ozarks is a high plateau, cut a few hundred feet many times by creeks and rivers. It is having to cross so many of these valleys that makes southern Missouri seem mountainous. When you finally come out on top of the plateau, north of the Missouri river, the Ozark Plateau is about as flat as Iowa.
