Reweighted score voting
Each voter submits a ballot which, for each candidate, indicates a numeric score which is less than or equal to some maximum number MAX.
Each ballot is given an initial "weight" of 1.
Repeat the following P times, where P is the number of winners to be chosen:
1. The weighted scores on the ballots are summed for each candidate, thus obtaining that candidate's total score.
2. The candidate with the highest total score (who has not already won), is declared a winner.
3. When a voter "gets her way" in the sense that a candidate she rated highly wins, her ballot weight is reduced so that she has less influence on later choices of winners. To accomplish that, each ballot is given a new weight = 1/(1+SUM/MAX), where SUM is the sum of the scores that ballot gives to the winners-so-far (if one person won multiple seats, they contribute multiple times to SUM).
See http://www.rangevoting.org/RRV.html for more details (some of the wording on this page is taken from there). A variant is to use the reweighting formula 0.5/(0.5 + SUM/MAX).