Something I'm surprised no-one has mentioned - in addition to saying clearly and plainly why you can't shake hands, replace the handshake with a different, equally respectful physical gesture.
For example, a bow.
(not a big bow from the waist, which could seem embarrassingly theatric - a subtle but noticeable dipping of the head and shoulders in their direction will be understood as respect)
This is important so the other person isn't left hanging, feeling foolish or rejected because their gesture wasn't reciprocated. Reciprocate the gesture, but differently. Then explain why.
To give an extreme parallel - in countries hit by the recent West African ebola outbreak, shaking hands was considered unadvisable. What succeeded was replacing the handshake with a non-contact physical gesture (they went with placing a fist over the heart, like a solidarity gesture). Asking people to do no physical gesture at all when greeting was not very successful - it felt very cold.
Since people will be surprised that you didn't shake their hand, I'd suggest also going out of your way to also do other things which communicate warmth, good intentions and attentiveness, to minimise any lingering awkwardness.
Say the person's name, make eye contact, smile warmly. And explicitly say that you look forward to working with them. You want to make it absolutely, explicitly clear that you're very happy and comfortable to work together.
[Smile, nod towards the extended hand so you acknowledge it, then make eye contact] I'm very pleased to meet you, Susan. [Bow slightly]
[If they look surprised or confused - smile warmly and say something like] I'm sorry, my religion only allows me to shake my wife's hand. But I look forward to working with you.
[If they don't look surprised or confused, make sure you take a minute to explain quietly later on, in case they are simply good at hiding such things]
Note how I worded the explanation in such a way that it comes across as, 'this is something reserved for my wife', not 'this is something not for people like you' - the positive side, not the negative side. Pro-wife not anti-women. (obviously, 'future wife' if you're unmarried).
It'll be obvious that the rule only applies to women from context, there's no need to labour the point. The key things to communicate are:
- that you are reciprocating their friendly greeting
- that it's religious not personal
- that it's about reserving something for marriage (so it can't be misinterpreted as being anti-women)
- that it only applies to this one thing (handshakes), and that every other aspect of your working relationship is business as usual