From Virginia:

Take I-81 South to I -40 East to Exit 435 (Newport) and follow Hwy 321 South all the way into Gatlinburg. Approximate drive time from interstate: 45 minutes

From Lexington:

Take I-75 South to I-640 East, then I-40 East to exit 435 (Newport) and follow Highway 321 South to Gatlinburg. Approximate drive time from interstate: 45 minutes
 

From Chattanooga:

Take Exit 81 on I-75 North and follow Highway 321 North through Maryville and Townsend and Highway 73 to Gatlinburg. Approximate driving time from interstate: 70 minutes.

From Nashville:

From I-40 take I-75 South to Exit 81 and follow Highway 321 North through Maryville and Townsend and Highway 73 to Gatlinburg. Approximate driving time from interstate: 70 minutes.

From Asheville:

Take I-40 West to Exit 443 and follow Foothills Parkway to Highway 321 South to Gatlinburg. Approximate drive time from interstate: 45 minutes.

From Atlanta:

Take I-85 North to I-985 North to US 23. Take US 23 to Highway 23/441 to Highway 74/441 through Cherokee, NC to Gatlinburg. Approximate drive time from Atlanta: 4 hours 10 minutes.

Click here to view the alternate routes map

This information gathered from the Gatlinburg Chamber of Commerce.