I think I heard somewhere that a person could not multiclass Rogue/UnchainedMonk, because you'd have to do either Rogue/Monk or UnchainedRogue/UnchainedMonk.
Is this right? And if so, where does it say this rule?
That is incorrect, you can be a rogue/unchained monk.
It appears whoever told you that has misinterpreted the line "These classes can be used alongside their original counterparts (although individual characters must use one version or the other exclusively)." which is only meant to stop characters from being things like a rogue/unchained rogue.
Milo's got it. Unchained monk is an alternate class of monk (like samurai, antipaladin, etc), so those two don't multiclass.