One alternative is using cards to assign Euchre partners. This is a good option if you have 9 tables or less or want to assign the Jack through the Ace as table labels. You have 2 red cards and 2 black cards for each table. You can lay the cards face down and have players chose the cards to determine their partner for each game. The two players with the red cards with the same number are partners at that table and the two players with the black cards with the same number are partners. One of the groups I attended used Giant Playing Cards so the cards to assign partners would not get confused with the cards used for playing. Using cards is an easy alternative.
Another alternative is poker chips. For each table, you have 4 chips: 2 red and 2 blue with the table number on the chips. Players chose a chip for each game and the two players with the red chips with the same table number are partners and the two players with the blue chips with the same table number are partners. Since chips are more durable than cards, this is a good alternative for a group that meets regularly.
Another alternative for rotating Euchre partners for a VERY large group is rotating by tables. There are different ways of doing this, but this method seems the easiest to me. With a large group, everyone chooses a spot to play the first game. At the end of each game, the partners with the least points (the losers) stay at that table and become opponents for the next game. The partners with the most points (the winners) get up and move (winners "walk" is an easy hint of who moves to the next table) and then become opponents for the next game. If there is a tie score, each team can draw a card and the team with the highest card would move to the next table. Before the winners move to the next table, each partner picks a card. So there will be a Winner-high card and a Winner-low card player and a Loser-high card and a Loser-low card player. When the winners move to the next table the Winner-high card will be partners with the Loser-high card player and the Winner-low card will be partners with the Loser-low card player. In this method, it is important that the tables are numbered and in order so the teams can easily move to the next table. (Image from http://www.ca.class.posot.com)
If the number of players is not a multiple of 4, there will be players that will have to sit out some games. In most of my groups we deal around 8 times per game and players receive 7 points when they sit out. To assign sit outs, have a card or chip which says "sit out" if you are using one of these alternatives to assign partners. If the teams are moving tables, the players coming from the highest number table will become sit outs and the sit outs will re-enter the game at Table 1.