B737 Captain reported uncontrollable oscillation in flight followed by engine vibrations. An air turn back and precautionary landing were made at the departure airport.

Narrative

Passing 10;000 ft. on climb from ZZZ; we accelerated to programmed climb speed of 319 KIAS and pitched to an attitude that produced a 4;000 FPM rate of climb. The aircraft had 26 Passengers onboard and was pretty light with fuel. As airspeed passed 300 KIAS; the aircraft began to oscillate in an up/down motion. Very steady and cyclic. Each movement was identical. It felt as though we were passing over very symmetrical chop on water. At first we thought it was turbulence but realized this was not the case due to the uniformity of the motion. After an assessment and some conversation between the two of us; we thought we could have a problem with the aircraft. We reduced airspeed to 280 KIAS and pitch to 1;000 FPM. When the aircraft decelerated through 290 KIAS; the movement ceased. Engine vibration showed no more than 1.7 and was less when power was increased to try and continue climb. At this point the oscillation began again and was just as steady as the first time. No vibration was felt through this entire event. Just the steady oscillation/up & down bumping. We both agreed that we were uncomfortable continuing to ZZZ1 as a serious airframe or flight control problem could exist. We made the decision and requested priority handling with ZZZ Center; and they gave us expedited handling back to ZZZ. We also notified Dispatch via ACARS; and they provided us with landing data that we cross referenced with performance data. No adverse issues were encountered during descent and uneventful landing at ZZZ. Maintenance met aircraft at Gate; and we briefed them on what the problem was. Logbook write up was made by me; and Maintence verified the write up had the information they needed while we all discussed the issue in the jet way. Aircraft was towed off the gate and taken to hangar; and we continued on to ZZZ1 in a replacement aircraft.