Icelandair and PLAY have round-trip flights from New York (JFK) to Reykjavik (KEF) for $238-$361, NONSTOP. Flights March to June. Extended weekends available.
A Carry on bag (20-lbs @ 21.6 x 15.7 x 7.8 inches) and a personal item (15.7 x 11.8 x 5.9 inches) is allowed for free on Icelandair. You can avoid seat selection fees by accepting the randomly assigned seat at the gate.
Icelandair offers free stopovers, which good if you want spend a few nights in Reykjavik. More details are on their website (link).
PLAY is a budget carrier that has more add-on fees for basic amenities.