In case anyone finds it useful, www.querymongo.com does translation between SQL and MongoDB, including OR clauses. It can be really helpful for figuring out syntax when you know the SQL equivalent.
In the case of OR statements, it looks like this
SQL:
SELECT * FROM collection WHERE columnA = 3 OR columnB = 'string';
MongoDB:
db.collection.find({
"$or": [{
"columnA": 3
}, {
"columnB": "string"
}]
});