B737 NG Captain reported encountering wake turbulence while in the trail of a B747 during a descent.

Narrative

During the descent on the LINKK1 we were in trail of a heavy 747. They - 747 - were cleared to descend via in front of us approximately 10 miles ahead. The winds aloft were mostly a headwind around 30 - 40 kt. The flight was smooth across the Gulf of Mexico; seatbelt sign was off. During the descent we encountered a light turbulent airframe vibration lasting 15 - 30 seconds; followed by the yoke displacing 30+ degrees to the right; at which point the autopilot disconnected and we got an oral roll authority message. The First Officer observed the yoke movement; and took control as soon as the autopilot disconnected. The aircraft rolled slightly to the left. The First Officer took control of the aircraft and turned 10 to 15 left into the wind to escape the wake turbulence. I; the pilot monitoring; and Captain notified ATC of the wake turbulence and told them we would need to offset 1 to 2 miles behind the heavy. I immediately turned on the seatbelt sign and made a PA. I checked with the flight attendants after we were clear of the wake. No issues in the cabin were noted. During cruise we observed the heavy 747 contrail ahead of us; and I relayed a story of a previous wake turbulence encounter in light winds; so we had some anticipation of the wake effects and responded immediately and appropriately.