There used to be a classic thread on this topic, but apparently it was deleted--no idea why. I can't hope to recreate the language of the original, but it's certainly time--now that it's February and some teams are already working on the sprint relay--to restate the basic points.
1. All else being equal, a 4x100 relay squad runs faster if each member keeps the baton in one hand throughout her/his individual leg.
2. The first and third legs are run around curves; those runners will have faster carries if they stay right next to their inner lane line.
3. If *all* runners hug the inner lane line, there's a good chance that someone coming in to make a pass will run up on the outgoing runner, with a trip and/or a dropped baton real possibilities.
4. So it makes sense to have the second and fourth runners stay in the outer parts of their lanes, which are straightaways anyway.
5. Hence the first and third runners should carry the baton in their right hands, and the second and fourth should carry it in their left. That way, the baton can be passed "straight ahead," yet the runners are in no danger of running into each other.
6. An easy way to remember all this: The baton stays in the middle of the lane. The leadoff runner wants to hug the curve, but the baton stays in the middle of the lane, so that first runner carries it in the...yes, right hand, and passes it to the left hand of the second runner. Then #2 passes it from her/his left hand to the right hand of #3, who stays in the inner part of her/his lane and passes the baton to the left hand of the anchor runner.
I've been told that all of this is common sense and doesn't even have to be said. Well, as the man said: common sense is like common courtesy--not that common. Every year I have seen good teams screw this up, including Olympic squads. Keep it simple: The baton stays in the middle of the lane.