Seating Arrangement Problem

[cabiness42]

This has a poker application but really is a math problem, so I post it here.

8 people want to play a series of 8 SNG tournaments.

Instead of randomly drawing for seats before each tournament, they want a system whereby each player is guaranteed to begin each tournament in each position.

The simple solution is that you seat all 8 players in the same order each time and start the button in a different place each time.

However, this results in having the same players on your left and right for each tournament, which is not fair given the differing skill levels of the players.

So, is there a mathematical way to ensure that every player starts from every position, yet doesn't have the same players on his/her right each time?