Is it important to put points in piety for Alchemists, Psions and mages? The description for piety leads me to believe that it'll affect my mana points even for non-priests. Is this true?

Yes. Although piety primarily affects priestly spells, it also adds spellpoints to a caster's total point allotment. Finally, several folks *think* that piety may have an effect on how swiftly a character upgrades known skills. I tentatively agree with this.

Piety directly affects spell points in each realm. Increase your piety during level up and observe. Of all the things you do to grow SPs, Piety has the strongest affect.

Piety also leads to IronWill, which adds resistance to spells (something the low HP professions, like mages, will find useful).

Piety affects Divinity by making it 'easier' to learn. (PCs w/high divinity grow divinity skill faster than those w/low stat.) A few other skills also use piety, but I think the effect can be safely ignored for non-Priests.

Piety also affects HPs, Carrying Capacity, and Stamina, about half as much as Vitality and Strength do.

I recommend that mages focus on INT/PIE or INT/(PIE/SPD). Wizards run out of SPs too fast. They're also handicapped by low HP, CC, and Stamina.