I'm answering this without reading all the other responses knowing I may be embarrassed by a much better response in a few more pages from where I read, lol!
If I understand it correctly, we have two different 'knowns' going on with the pings. There is the time 'known', through which the arcs were figured. That gives a distance from the satellite that is equidistant, yes.
But, if there is a doppler effect, there are also wavelengths (of sound in the cases I've been reading about). But wavelengths aren't time, so I take it it's fair to say that's the second 'known': the wavelengths.
The satellite really was either north or south of the plane, so the constraint of the "distance only" data was just a data problem. If they had a way to measure the appropriate wavelengths to see a doppler effect, that could tell direction.
Presumably all the data crunching was comparing known entities (other planes, maybe quick studies, etc) to see if the doppler effect they were able to pick up was significant enough to actually mean something. And I guess it did!
A few analysts' blogs I've been reading have been hypothesizing a 'subtle doppler effect' for a while now, so I trust that it's possible (and I trust Inmersat that way anyway, them being good ole careful Brits

).