How can I power an Arduino in a 3-phase environment?
The very short answer is that the Arduino board itself is designed to be powered by low-voltage DC (but you almost certainly already know this). So what you're really asking, I think, is "can you use the mains power supply that converts mains AC to DC on a three-phase supply?"

The answer to that is a guarded yes. You will need to confirm that you are connecting to a single phase (basically what you were doing in testing) and that the voltage is appropriate for your power adapter.

The abridged version of the gory details follows…

Three-phase supplies are almost always used for motor or other power applications – for a low power device like an Arduino (or even lighting or small appliances) you don't typically need three-phase. What is usually done is to take the power supply for those devices either from phase-to-neutral or phase-to-phase – depending on the configuration of the three-phase supply.

Wikipedia has an article on three-phase power that explains the common connection schemes (Y or star where the three phases are wired to a common point, and delta where they are wired in a triangle). What is significant is that you will typically get a high and a low voltage depending on how you connect to a three-phase system. For example in the US a common Y three-phase service will give you 120 V phase-to-neutral and 208 V phase-to-phase.

A true 3-phase (as in needs to connect to all three phases) low-voltage power supply will probably be quite expensive. But you also don't need one. Unless you are building an "industrial" device all you need is a "normal" low-voltage power supply that is designed for the voltage that you'll be connecting it to.

All you need to do is to confirm the voltage that you'll be using and check that your power supply is rated for it. Many low voltage supplies are designed to be "universal" and you'll see a rating like 100-240 V, 50-60 Hz. That's likely to work on most three-phase supplies as long as the connection isn't phase-to-phase in a part of the world where the high voltage is in the 400 V range.