Unless you've got language skills and/or some connection to a "really" foreign country (anywhere cozy with the U.S. doesn't count), it'll be easier to maintain a reasonable standard of living anonymously in the U.S. than in a foreign country.
Assuming you don't have boat loads of cash, you're going to need to start some sort of small business/self-employment that can stay beneath the radar (i.e., mostly cash, preferably mobile). It's easier to do this in some states/counties than others.
Landscaping, handyman, and maid service are fairly obvious. Another idea for the crowd here is mobile mechanic. All you need for that is a vehicle, tools, and a pre-paid cell phone. The biggest problem is people wanting to pay you with credit cards, but you can probably work around that. Having some sort of lunch wagon for job sites is another option, but the permitting requirements here can catch up with you.
The other alternative is jobs where people move around a lot, but you'll need one or more fake identities that will pass muster. Things like construction worker, waiter, etc. The problem with most of these is that they don't pay very well.
For living, your best bet is some sort of RV. There are so many campgrounds (and Wal Marts) that you can easily pay cash and move around every few months without arousing suspicion. Pretty much any other place you can live that paying cash won't arouse suspicion is going to be a bit of a dump.
You're going to want to stay in a reasonably metropolitan area. It's easier for new people to blend in and easier to remain anonymous in such an environment.
Fake identities are easy enough to get for the odd thing that needs it, but making a habit of using a fake identity in the U.S. will eventually get you caught.
Most of the other ideas here are pretty good, the main point being leave no connection between your old and new life. Every single link is going to dramatically increase your odds of getting caught.