It appears that the Tu-144 was used in passenger service for a short period - until the second crash:
The Tu-144S went into service on December 26, 1975, flying mail and freight between Moscow and Alma-Ata in preparation for passenger services, which commenced in November 1977 and ran a semi-scheduled service until the first Tu-144D experienced an in-flight failure during a pre-delivery test flight, and crash-landed with crew fatalities on May 23, 1978. The Aeroflot flight on June 1, 1978 would be the Tu-144's 55th and last scheduled passenger service.
A scheduled Aeroflot freight-only service recommenced using the new production variant Tu-144D aircraft on 23 June 1979, including longer routes from Moscow to Khabarovsk made possible by the more efficient RD-36-51 engines used in the Tu-144D version. Including the 55 passenger flights, there were 102 scheduled Aeroflot flights before retirement of commercial services.
It is known that Aeroflot still continued to fly the Tu-144D after the official end of service, with some additional non-scheduled flights through the 1980s. One last report showed that it was used on a flight from the Crimea to Kiev in 1987.
http://www.wordiq.com/definition/Tupolev_Tu-144
Edited to add a link to a proposed Tu-144D routemap that envisioned Havana, Tokyo and New York flights:
http://www.tupolev.ru/English/Picture.asp?PubID=2390