A few points to make here.
As you noted, the Greek here is a bit slightly ambiguous and could go either way. For the purpose of your question, we are assuming a particular reading, so I will avoid that discussion here.
But let's back up a bit, and see the whole quote:
On the last day of the feast, the great day, Jesus stood up and cried out, “If anyone thirsts, let him come to me and drink. Whoever believes in me, as the Scripture has said, ‘Out of his heart will flow rivers of living water.’”
(John 7:37-38 ESV)
Consider that ancient Greek has no punctuation, and take this quote as a whole. Parsing it differently, you can see this rendered
“If anyone thirsts, let him come to me and drink, whoever believes in me, as the Scripture has said, ‘Out of his heart will flow rivers of living water.’”
That is, "whoever" is referring to him who thirsts ("if anyone thirsts, let him come"), and the water is coming from Jesus ("let him come to me and drink"). Then the Scripture is not saying water will come from the believer but instead from Jesus.
And, indeed this seems more plausible. Several passages teach similarly, but before looking at them more closely, it's necessary to take a step back further yet.
Verse 37 refers to a feast, but one has to go all the way back to verse 2 to see that this is referring to the Feast of Booths (a.k.a Feast of Tabernacles). Though this feast was instituted in Leviticus, it was more often than not honored in the breach. One instance when it was celebrated, however, was during the rebuilding of Jerusalem under Ezra and Nehemiah.
Beginning in Nehemiah 8:13 and continuing through Nehemiah 9, we see the Feast of Booths celebration taking place, and those present honored God by remembering what they had done for him. Several times in this prayer, the priests remembered God's provision for them from the rock that Moses struck to brought forth water (see Exodus 17:6):
You gave them bread from heaven for their hunger and brought water for them out of the rock for their thirst, and you told them to go in to possess the land that you had sworn to give them.
(Nehemiah 9:15 ESV)
So then, flowing water has been connected to the Feast of Booths before; and this is not the last word that Scripture says about this rock from which water sprang either--in 1 Cor. 10:4, Paul explains it always was from Christ that the water sprung:
For they drank from the spiritual Rock that followed them, and the Rock was Christ.
So then, Christ to be the source of water flowing for the thirsty is no new thing at all--he was the source for Israelites in the Exodus as he is for all who believe in him.
This theme of water flows throughout the Bible, including those passages mentioned by @Richard. He has already dealt with those, so I will not rehash that specifically, except to note that (as Richard did) these verses refer to the source of living water as God, which is more consistent with understanding the source in John 7:38 to be Christ rather than the believer.
Additionally, as @JackDouglas has pointed out, Ezekiel 47 refers to water flowing from within the temple, which would be at least awkward if the source of water is the believer. Again, consider Jesus. If the temple is the meeting place between God and man, Jesus is the ultimate meeting place between God and man--the temple par excellence. It is natural for him to be equated with the temple, and the source of water in Ezekiel 47.