B737-700 Captain reported encountering wake turbulence in cruise flight at FL340 from a preceding B737 that resulted in a sudden 30 degree bank left and then right.

Narrative

We were cruising at FL340 with a clearance to descend via the SILCN6 Arrival; well prior to the TOD. I had noticed in the windscreen an aircraft ahead of us putting out contrails that were curling into perfectly symmetrical swirls but didn't think much about it. I had assumed it was 1;000 ft. or so above or below and was coming against us; but I believe now it was co-altitude company traffic going the same direction. We had a new hire pilot on the jump seat doing an observation ride and were talking flying/Company stuff which probably distracted me from possible hazard ahead. Fairly suddenly; the aircraft rolled 30 degrees to the left; and then started back rolling to the right. I immediately recognized it as wake turbulence and hit the altitude intervention button to start a descent to get out from it. The 1970's 286 processor took too long for me to start the descent; so I punched off the autopilot and started the descent manually. Once the electronics caught up; I reconnected the autopilot; probably within 500 ft. of descent. I asked my FO (First Officer) to report it to ATC whose reaction was utter disbelief by his tone and words used. That's when I learned it was Company ahead of us. I don't know if it was a larger version of the 737 or not. I did notice we had about 116 knots of tailwind directly at our 6:00 position probably exacerbating the setup (or at least not alleviating it). My FO looked up if it was a reportable incident or not; and the manual just stated ATC was all that was required. Our Flight Attendants (and therefore our customers as well) definitely noticed it; because they mentioned it on the ground. I decided to write this report to have as a data point; and because ATC was so utterly surprised and really non-believing that these things happen (and happened!). I don't remember how far we were from Company; but I am guessing within 20 NM? I think ATC mentioned the distance when we reported it; but I was busy trying to maintain aircraft control; so I didn't hear exactly what he said. I should have analyzed with TCAS how far ahead and what direction the aircraft was that was making the pretty contrails. Pilot crews (me included) and ATC should be more aggressive at putting in route offsets when the potential for wake turbulence exists. Educating our crews on our FMS's offset function could help in that effort. I don't think Controllers are all that aware of the hazard.