Starting with Dining Cryptographers networks (DC-net), several peer-to-peer (P2P) anonymous communication protocols have been proposed. Despite their strong anonymity guarantees none of those has been employed in practice so far: Most fail to simultaneously handle the crucial problems of slot collisions and malicious peers, while the remaining ones handle those with a significant increased latency (communication rounds) linear in the number of participating peers in the best case, and quadratic in the worst case. In this work, we conceptualize these P2P anonymous communication protocols as P2P mixing, and present a novel P2P mixing protocol, Fast-DC, that only requires constant (i.e., four) communication rounds in the best case, and 4 + 2f rounds in the worst case of f malicious peers. As every individual malicious peer can halt a protocol run by omitting its messages, with its worst-case linear-round complexity, we find Fast-DC to be an optimal P2P mixing solution. We find Fast-DC to be an ideal privacy-enhancing primitive for Bitcoin and other emerging P2P payments alternatives. Public verifiability of their pseudonymous transactions through publicly available ledgers (or blockchains) makes these systems highly vulnerable to a variety of linkability and deanonymization attacks. Fast-DC can allow pseudonymous users to make their transactions unlinkable to each other in a manner fully compatible with the existing systems. As representative examples, we extend Fast-DC to create unlinkable Bitcoin and Ripple path-based settlement transactions. We also demonstrate practicality of Fast-DC with a proof-ofconcept implementation, and find it to require less than 16 seconds even with 50 users in a realistic environment.