American Airlines, British Airways, Finnair, Iberia and various other airlines have round-trip flights from Seattle (SEA) to Larnaca (LCA) for $599 with a connection. Flights January to March. Extended weekends available.
There are cheaper fares with inferior routing on Lufthansa, those flights are about $581, but have a 30-hour connection time on return. Avoid those unless you really like spending time in transit.