Kimberly,

Number each team in a circular order (...13, 14, 15, 16, 1, 2...)

In the first two weeks, everybody plays the teams with adjacent numbers (16 is next to 1)

1,2; 3,4 ;5,6; 7,8; 9,10; 11,12; 13,14; 15,16

then

1,16; 3,2; 5,4; 7,6; 9,8; 11,10; 13,12; 15,14

In the next two weeks, everybody plays the teams two ahead and two behind.

1,3; 5,7; 9,11; 13,15; 2,4; 6,8; 10,12; 14,16

then

15,1; 3,5; 7,9; 11,13; 16,2; 4,6; 8,10; 12,14

In the next two weeks, everybody plays the teams with numbers off by three.

And so on, till in the last week everybody plays the team with a number differing by 8:

1,9; 2,10; 3,11; 4,12; 5,13; 6,14; 7,15; 8,16

You will see this will work for any even number of teams.

Good Hunting!
RD