I believe transactions-as-proof-of-stake (the heaviest subtree model) is probably the best alternative to proof-of-work - and it isn't all that good.

The basic idea is that the "finite resource" available for deciding to prefer one chain over another, is the set of unspent txouts that exist at the point of the chains' divergence from each other.  If transactions must give the hash of a very recent block (parent or grandparent to the block they'll be in) then they can be counted as a "vote" for a chain including that block. 

In practice, this makes it possible for an attacker to spend ten coins in one chain, then support a different chain by spending a thousand coins (probably to himself) there, and if the second chain is accepted it 'unspends' his ten coins.  Obviously this only works as a double spend if he does it before everybody else in the course of regular spending puts the first chain a thousand coins ahead. 

But it gets worse than that, because at any given moment there may be dozens or even hundreds of crooks looking for a chance to double spend, and if two competing chains appear, their efforts to make a small initial expenditure in the apparent leading chain and then dump a huge transaction into the second chain all reinforce each other. 

On one hand, if everybody understands the security requirement for transactions as proof of stake and regularly transacts their coins several times a day, (which you can arrange with a proof-of-stake interest/security payment for each transaction) the crooks shouldn't be able to overwhelm that traffic with their timing games.  On the other, that would generate an absolutely enormous blockchain and have a high communications overhead.

So in the short run, it doesn't work.  In the long run, it can provide an absolute security guarantee given enough time; Once more than half of all the coins in txouts that existed before a block was created have been spent, that block becomes absolutely irrevocable no matter what proof-of-work anybody pours on or what manipulations they do with spending and transactions.