Bram,

For some reason 12 is a hard number of golfers to get a good schedule for. I wrote a program and had it search through about 50 million of the $4 \times 10^{16}$ possible schedules (4 followed by 16 zeros). That took a couple of days. Here is the most balanced schedule it found. I suspect it isn't very good. The 12 players are the 12 positions, and the number in each position is the group (0, 1, or 2) that player is in on the given day. For example, on Day 1 the groups are 1, 2, 3, 4; 5, 6, 7, 8; 9, 10, 11, 12. Tee times can be assigned to the groups arbitrarily.

Day 1 : (0, 0, 0, 0, 1, 1, 1, 1, 2, 2, 2, 2)
Day 2 : (0, 1, 0, 1, 1, 1, 2, 2, 0, 2, 0, 2)
Day 3 : (0, 1, 1, 0, 1, 2, 0, 2, 2, 0, 2, 1)
Day 4 : (0, 1, 1, 2, 2, 1, 2, 0, 0, 1, 2, 0)
Day 5 : (0, 1, 2, 0, 2, 0, 2, 1, 2, 1, 1, 0)
Day 6 : (0, 1, 2, 2, 0, 0, 1, 2, 1, 0, 2, 1)

Can you do better by hand?

Victoria