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.