Movement is an odd topic.
My friend and I were talking about this not too long ago. He was looking for a movement system that offered "Freedom of movement". After a while I still wasn't completely sure what he meant by the statement.
To me movement requires a set of restrictions. 'You can do this, but you can not do that.' It really depends on the mechanics of your game.
For instance, Warhammer (and many other miniature games) use rulers to judge distances. This means that players can move in any direction (with exceptions to peices that have limitations of course). This does however cause other issues... Dealing with odd distances. Or margins of error.
People before me have done a good job outlining board layouts, so I don't feel the need to reitterate them.
All those systems impose rules by their very nature. Rules like, a peice can only be in the center of a square, or on the edge, or on a corner. Sometimes what you want to accomplish can be figured out by looking at the rules you need to impose on players.
Also, some games are easily adaptable to other systems. Open d20 for instance can easily be modified to a hex based movement, from a square based one (I prefer hex myself).
Also another type of board are the combination types, like octagons and squares alternating, or hexes and triangles. If you are good with object relationships I am sure you can come up with some odd designs.
Wow, that was dumb.
I mistakenly put this in root, instead of replying to 'Area Movement Question: Grid, Hex, or Something Else?'
(http://www.bgdf.com/modules.php?name=Forums&file=viewtopic&t=1491)
My bad... Not sure how to fix this...
Appologies to all.