There's always trade-offs in the voting system. If you go too far one way, like towards LastFootnote's system, you'll knock out the most objectionable cards, which can at some point include the best options. Eventually you're left with cards which nobody hates, but which aren't stellar either. Like, you end up with wishing wells or labs all over the place. Which is fine, but lacks some panache.
On the other hand, if you go too far towards a plurality system, you end up with a card that certainly has panache, but which can be really objectionable to lots of people, potentially even totally broken.
In any single-vote system, you have the problem that the card everyone likes second-best, which, if there are 20 different cards, is probably the best one, can easily get knocked out first. Not really sure about fixing this though, without introducing other problems.
Overall, other than the problem just above, TINAS's suggestion of the Alternative Vote system looks pretty good, except that it could easily take a long time? I guess if you just have people who voted for the eliminated thing change THEIR votes, you can speed it up.
Finally, you have the issue of public vs private voting. Public voting has problems in that you are disincentivized from voting until the last minute, in order to make your vote have the most potential impact. But private voting has the downside that you can easily have large numbers of people casting totally irrelevant votes, and you basically lose their valid opinions on the process. Of course, the public v private thing also has interplay with the voting system you choose.