How did planes measure their speed relative to the ground before GPS?
The other answers seem to make a simple thing way too complicated. Yes there are DMEs or TACANs, that are sometimes used to get a ground speed, but it's only accurate if you're tracking exactly to or away from it. And it certainly wasn't used to calculate speed enroute unless one of your navaids was a VOR/DME or equivalent. Rather than using DME they'd be using INS, which delivers data more similar to GPS.

Calculating ground speed in the past (before INS, or in airplanes without one) was done using the simple method of having checkpoints along your route. These checkpoints where navaids or intersections determined by multiple navaids. Either way their location was known, divide the distance by the time spent travelling between them and you got your ground speed.