This is exactly what I've been thinking about for a long time. To bypass all the stress and worry, I figured why not just open a frickin' foreign bank account (preferably non-interest bearing), use it for depositing to and from online casinos and wiring to my US bank account? Your US bank breathes easier that the wire is from a bank rather than some place that uses cryptic descriptions. If the US bank inquires about it, it is easier to explain: "I have an offshore bank account for investment possibilities; for diversification purposes; for foreign travel overseas..." It would be hard for them to disprove any of these explanations which they likely just needed one anyway to cover their own ass to meet bank regulations. The inconvenient thing is most, if not all, banks outside US would require a visit in person to open an account but I'm not an expert on this. I just don't know of a foreign bank that will let you just open an account online (and do it from the US.)
Banks with online account servicing do not require a personal visit. They will verify ID electronically, and may use similar measures to those used by online casinos. There are places that specifically serve the market for offshore banking. There are banks in Jersey, Guernsey, Luxembourg, even Switzerland. Governments worry about the TAX collection from these accounts, rather than what they are used for (provided this is legal under INTERNATIONAL law). Recently, there have been moves to get better cooperation from these countries for the purposes of cracking down on tax evasion.
They MAY ask for a personal visit, or justification, from a US citizen wanting to open an account. These offshore banks cater for the weathier people, so they will want to "smell the money" before deciding whether you can have an account. They are unlikely to be interested in someone putting in $100 for "investment and travel", but might bite if $50,000 was offered for purposes such as investment and travel.
Another reason to have a foreign bank account is to buy property in that country, in many cases, this is a REQUIREMENT. If you have the money, and are "looking to buy a property", you should find it easier to open an account.
UK citizens often buy property in Spain & France. This involves opening a "local" bank account, as well as keeping the main UK bank account. The local account is used for payments and receipts in the country where the property is to be bought. They may still require a visit in person though.