Below is a grid suitable for use in a 7-team league. You would run 7 games each week, on League Night.

To read the chart, run across a row for each team, or down a column for each game. So for example Team 1 plays on Red in game A, Yellow in game E, and Green in game G; and Game A consists of team 1 on Red, team 2 on Green, and team 4 on Yellow. (If your site has a blue team, substitute colours appropriately.)

7 teams in the tournament.
3 teams play in each game.
3 games are played by each team.
7 games are played in the whole tournament.
Each team plays every other team exactly once.
Each team plays on every colour exactly once.

Team1R Y G
3 GR Y
5 Y GR
6 Y GR
7 Y GR

To choose which is Team 1, which is Team 2, and so on, you can use whatever means of draw you like. One good method is a random draw during the pre-season captains' meeting, where the captains pick cards numbered 1 thru 7 from your hand.

After each week, you need to shuffle the team names for next week's games. Again, you can use whatever mechanism you like: you can pick randomly again, or you can renumber the teams according to a rule. Note that just bumping every team's number up by one is not a good rule: the same three teams would wind up playing each other together each week. The following would be a good rule: 1 to 3, 2 to 6, 3 to 2, 4 to 5, 5 to 1, 6 to 4, 7 to 7.

If you want to print one of these grids, be aware that Internet Explorer's printing abilities for tables (especially wide tables) are pretty lame. So if you're using IE, then try Select All, then Copy, then open Excel, and go Paste Special, HTML. It's still pretty lame, but not as lame.