Hmm, after some more reading and googling, it looks like the following will work:
GET index/_search
{
"query" : {
"bool" : {
"should" : [
{"terms" : { "user" : ["john", "jack", "bill"]}},
{"terms" : { "fullname" : ["john", "jack", "bill"]}},
{"terms" : { "firstname" : ["john", "jack", "bill"]}}
]
}
}
}