Is there an easy way to convert rows in BigQuery to JSON?

This is now possible using TO_JSON_STRING, which can also help when you want to pass a row from your table to a JavaScript UDF.

#standardSQL
WITH MyTable AS (
  SELECT 1 AS x, 'foo' AS y, true AS z UNION ALL
  SELECT 2, 'bar', false
)
SELECT TO_JSON_STRING(t) AS json
FROM MyTable AS t;