This article is from the Golf FAQ, by marcelo@nntpserver.Princeton.EDU (Marcelo A. Gallardo) with numerous contributions by others.
The next best alternative, according to USGA studies, is to use a hole
score selection system, often called the "Peoria system." Under this
system, a player learns his handicap after the round is completed. By
this method, the committee secretly selects a par-3 hole, a par-5 hole
and four par-4 holes from an 18-hole course. The par-4s should be
representative in length and difficulty with two chosen from the front
nine and two from the back nine.
A modified Peoria handicap is calculated by adding the player's strokes
over par on the six selected holes, and multiplying by 2.8. This will
equal the player's allowance to be deducted from his gross score. The
maximum hole score for allowance purposes is three over par on par-3s
and 4s and four over par on par-5s.