For his translations he has won the F. R. Scott Translation Prize, the Richard Wilbur Translation Prize, the American-Scandinavian Foundation Translation Prize (twice), the Inger Sjöberg Translation Prize, the Lewis Galantière Award, and the Harold Morton Landon Translation Award.

Greenwald has published three books of poems, Connecting Flight, Slow Mountain Train, and The Half-Life, and several volumes of poetry in translation, including North in the World: Selected Poems of Rolf Jacobsen (University of Chicago Press, 2002, paperback 2015), Guarding the Air: Selected Poems of Gunnar Harding (Black Widow Press, 2014), and Through Naked Branches: Selected Poems of Tarjei Vesaas (Revised Edition, Black Widow Press, 2018).

He has also translated a novel by Erland Josephson, A Story about Mr. Silberstein (Hydra Books / Northwestern University Press, 1995, paperback 2001), and a novel for young adults, I Miss You, I Miss You! by Peter Pohl and Kinna Gieth (R & S, 1999). He has published in numerous journals and has given readings in Canada, the U.S., England, The Netherlands, and Norway.

From 1970 until 1995 Greenwald edited WRIT Magazine, an international literary annual that he founded. He has delivered lectures on poetry at the University of Chicago and the University of Agder (Kristiansand, Norway) and lectures on translation at Boston University, the University of Oslo, and York University (Toronto).