It's important to state first that different pets will react differently to scents so it is a good idea to do a bit of research on your own animal and breed. Here are some general cautious measures:
1) Avoid certain oils around dogs especially : clove, tea tree, armoise, thyme, hyssop, clove, cinnamon are all very powerful and can have adverse effects.
2) Do not apply the oils pure near/on a pet (this apply to humans as well by the way). When sprayed in the air, up to 3-5% of essential oils would be OK but for topical applications, 1% is enough.
3) test how they react to the smell of a particular oil : you can do a sniff test by adding 5 drops of oil in a glass of water for instance.
And remember it's not a good idea to use a strong scent in a close environment. But if used only lightly in a room mist or a diffuser, essential oils will be fine. There are many more odours in their environment that could be more harmful. Just have a sensible approach to it.
And here are by the way several applications where oils are commonly used around pets: