B737 Undifferentiated or Other Model
Flight Crew Air Transport Pilot (ATP)
Flight Crew Type 13976
Flight Deck / Cabin / Aircraft Event Smoke / Fire / Fumes / Odor
Air Traffic Control Provided Assistance; Flight Crew Landed in Emergency Condition
Aircraft
About 45 minutes into the flight our flight attendants reported they smelled an odor possibly like burning electrical wires; but they also said some passengers reported a burnt rubber smell and one reported an odor like a burning clutch. We were cruising at around FL350. The odor seemed to be originating in the aft; right cabin area but the exact location could not be determined; and the odor was evident in varying degrees throughout the cabin. However; the odor dissipated rapidly; before any checklists were run. An ACARS report was sent to Maintenance Control. I asked the flight attendants to remain vigilant and report if anyone smelled this again. In the latter portion of the descent; I believe around FL250 or lower; they again reported similar odors; including a sulfur smell; and this time the odors were not dissipating. I ran the Smoke; Fire or Fumes checklist and [advised ATC]. Flight attendants were briefed. We were given traffic priority for landing in ZZZ; which was the nearest airport. As the checklist was run and switches were being turned off; the flight attendants reported the odor was diminishing but not very quickly. The odors were not reported as harmful so did not attempt to run a Smoke or Fumes Removal checklist. Because the time to landing was so short I was only able to accomplish the first 20 steps of the Smoke; Fire or Fumes checklist. The visual approach and landing to Runway XX was uneventful. After having the fire department inspect the outside of the aircraft just off the runway; where they saw no unusual indications; we continued taxi to the gate. A post flight exterior inspection by the FO confirmed nothing unusual. A conference call with the Dispatcher and Operations Manager was made after arrival at the gate.In retrospect; the FO and I considered the fact that ATC had issued a climb reversal after climbing through FL350 towards FL370. We had gotten to about FL353 then went back down. About five to ten minutes later we had gotten the first notice of an odor. The second notice was also associated with a descent; our final descent; which lasted much longer and had more significant lasting odors. Thoughts are this could be engine related; possibly something in the pneumatic system; that only presented itself when the engines were retarded towards idle.
B737 Captain reported a burning odor in the cabin that was more prevalent when the throttles were near idle.