I just read on another forum that a Monk will lose his unarmed attacks BAB if he multiclasses to another class. Is this true?

IIRC, monks can only multiclass to certain classes allowed by their order, though I can't recall them just off the top of my head.

While other classes get an additional attack at every +5BAB, Monks get the additional attack at +3BAB.
If a Monk gets even 1BAB from any class other than Monk, he looses this advantage.

Consider an L16 Monk with attacks at BAB 12/9/6/3.
Now however, if he took a fighter level his attacks would be 13/8/3.

This is NOT a bug.

Okay...

What if you take, for example, a single level of Cleric (+0 BAB). Would you be okay then?

Yes, that should work.

Afaik it will work so long as you do not get +1BAB from any one source other than Monk.

That's good. I was really planning on having my Monk take 1 level of Cleric.