Tracing Bitcoins to their original blocks Can I find (always, without fault) the specific set of blocks from which the Bitcoins that I hold originated?

If yes, would such a trace-back be computationally costly?
Yes you can, every transaction in the bitcoin network is registered in a public ledger called blockchain. You can query this ledger via http interface like https://blockchain.info/api/blockchain_api

I don't think will be costly but it depends how many hands your bitcoin passed. When you find transaction without a sender, you have to stop, this are mined (generated) blocks