Are there alternatives to Bitcoin which take a different approach? Are there any alternative approaches to decentralized, electronic currencies?

Bitcoin was built with certain technical characteristics in mind. I'm hoping to see if there are alternative approaches. I don't mean things like Namecoin or other adaptations of the blockchain concept.
There are "digital bearer certificate" systems, which are an older concept.  They require a centralized issuer in order to prevent double-spending.  You can think of the certificates as electronic IOUs for a commodity (such as gold) held by the issuer.

Some DBC systems implement Chaumian blinding to anonymize transactions.

Open Transactions is a recent DBC system that has gained some interest.  It could be used with Bitcoin as a backing commodity, taking advantage of the properties of both systems.