Thankfully I don't have this issue - I only employ people that fit in well with our culture and therefore, I or my fellow team mates, don't need to worry about avoiding each other.
This is obviously different in larger companies. I would suggest though that if someone upsets you just don't talk to them. But, even if you want to avoid someone, try to recognise if they're actually good at there job - they may be useful.
Remember, always be professional :).