Are there any natural supplements with anti-inflammatory properties but no bad side effects? If so, how do I go about choosing one that works/is safe?

No free lunch. There's no such thing as a free lunch in medicine. In other words, for any desired benefit from drug or surgery, there's always potential risk but it's not a guarantee of side effect either. OTOH, dietary supplements aren't required to warn you & thus tend to hype & mislead. Studies suggest benefit from turmeric, green tea, ginger, rosemary, cat's claw, devil's claw & willow bark (aspirin). Check.