How to move money between U.S and non-U.S bank accounts?

I may be moving to Switzerland soon and would like to know if there's a similar system to move money between a Swiss bank account and a U.S bank account.

There is no easy way. The most common method is International Wire or SWIFT. These kinds of transfer are generally charged in the range of USD 20 to USD 50 per transfer. It generally takes 2 to 5 days to move the money. Some Banks have not yet given the facility to initiate a International Wire from Internet banking platforms. One has to physically walk-in. So if this is going to be frequent, make sure both your banks offer this.

As the volume between US and Switzerland is less, there may not be any dedicated remittance service providers [these are generally low cost].

I have taken to using the service TransferWise. I have found them to be faster and cheaper and easier than using SWIFT, given the US Banking's... antiquated system of doing things.

I've made dozens of transfers between my international accounts with TransferWise over the past 18 months. Some of them very large and some of them tiny, and even when there's been an issue (I once wrote an offensive joke in my narration for the transfer and they noticed) they have handled it respectfully and quickly.

Prior to transferring money to US accounts, I used the SWIFT system - but SWIFT has a pretty spotty record in the US. Some banks you can do it all online but other banks, as Dheer mentioned, you have to go into the bank and sometimes find a senior staff member before you can find someone who even knows what SWIFT is.

Some brokerages will let you withdraw/deposit in multiple jurisdictions. e.g. I used to use Interactive Brokers. I could deposit/withdraw to US and UK bank accounts, in the appropriate currencies. It helps to have a brokerage that provides good rates on forex exchange also, and they were very good on the bid/ask spread. It was possible to get interbank rates plus a very low commission.

If possible, use HSBC (the bank). Get an account in each country with a local branch. Move your money between accounts fee free. Transferwise is a great service, but moving over a thousand bucks costs me more than $50. It's easy, but really not that cheap for regular money movement.