The board size and shape, and context of other pieces and rules surely have an effect on piece values. Large board make sliders more valuable w.r.t. to short-range leapers. (E.g. in FIDE B=N, while in Capablanca B-N = 1/2 P.) Small boards make long-rnge leapers virtually worthless (e.g. a Camel on 8x8). Wide boards (cylinder boards) make Bishops more valuable compared to Rooks.

The type of Pawns could have a large effect on piece values, as the dominant way in which Chess games are decided is in end-games where pieces help to promote, or try to stop promoting Pawns. I heve never attempted to measure this, but it would be very interesting to see how piece values change in Berolina Chess. To a lesser extent King type could affect piece value, as it certainly affects mating potential. In Knightmate Rooks have no mating potential, and this could very well suppress their value somewhet compared to the Bishop.

This is why I always report board size and context when I am quoting piece values.

To Mats: 

I am not denying the usefulness of Zillions at all. I am denying the uselfulness of the opinion of someone who loses to Zillions. If someone has a brilliant new strategy, which makes a Commoner worth more than a Knight, and using that strategy against Zillions he is clobbered by it, I would not see any merit in that strategy, and disbelieve the conclusion that it makes the piece as valuable as claimed.

I have some doubt if Zillions would rank above 1500 Elo in FIDE. In my estimate, it would hardly rank 1500 on a computer rating list. But computer lists might very well be somewhat expanded, as computers are on average much more similar than Humans, and it is well known that the more similar two programs are, the more extreme their mutual results are for the same perfrmance difference against a more varied set of opponents. So low computer ratings could very well be under-estimates compared to the Human rating scale, while the ratings of the top-engines (like 3300 for Rybka 4) would be over-estimates.

Zillions is incomparibly weak compared to Rybka in FIDE, but who isn't? I don't think you have to be as strong as Rybka to draw useful conclusions about FIDE piece values. Being a mere GM would already do. But a GM would crush Zillions at Chess, I am quite sure of that. 1900-rated club payers already have the upper hand against Fairy-Max at long time controls (although not at blitz, which is always more dfficult for Humans), and Fairy-Max is quite a bit stronger than Zillions in any variant it plays. (I hope you agree with that...) If Betza lost to Zillions with his Commoners, either he was very far from GM level in this variant, or the Commoners must have been so inferior to Zillion's Knights that even GM-level play could not avert the loss. In both cases the conclusion that the Commoners are better is suspect.

I do agree that Commoners get relatively better in the end-game, and I am sure that one can design posiions where a Commoner beats a Knight (or a Bishop, or a Rook), with the proper Pawn constellation. But on the averaege, I did not see any advantage of Commoners ove Knights, even in the late end-game. And I doubt if Zillions will (although this would not really be conclusive if a stronger engine does not).