Not too long ago I, too, was a math major utterly unprepared for getting a career. My strategy was to look for job postings with the word "Analyst" in the title, and focus on small companies. Those are the positions where you'll be working on solving innovative problems, at varying levels of definedness. Small companies, especially in niche markets, are more willing to hire people with a BS and good potential but not a lot of experience. Companies in a more niche or botique line of work, or a new emerging field, are also more willing to hiring people with more potential than experience (I got into web analytics myself, which hasn't been around very long).

If you're not sure about the opportunities you have, find a part-time position or a paid internship. It's less risk for you, and it's less risk for the company so it's easier to get the position. If it's something you like you can upgrade to full-time (as was my case), and if not, you can still gain skills and experience while shopping around.

In terms of lowering your expectations, you likely won't achieve the most prestigious position, but you can certainly find a position that's interesting and personally rewarding in an area that you want to be in. It's up to you to decide if that's lowering your expectations (I wouldn't consider it so).

No, don't lower your expectations. If you want to get into programming, you can. It's not clear how much you've done already, but what you don't know you can teach yourself. First, start some programming projects on your own or contribute to existing free software projects. That'll force you to learn most of the practical skills you need. If you want jobs where you get to work on interesting problems, you should start learning computer science theory too. With your background you should find it fairly easy to pick up much of an undergraduate CS education from a few good books. (In your position I'd start with SICP, which is available online with video lectures by the authors at MIT OpenCourseWare.)

Even before that, you can apply for entry-level software development jobs. Companies that hire new graduates don't really expect them to have useful real-world experience. (It's a big plus if they do, but it's not necessary.) If you can learn basic data structures and algorithms, for example, you might well qualify as a college hire at a big shop like Google/Amazon/Yahoo/Microsoft that has the resources to hire a lot of junior programmers and train them on the job. And use your friends, professors, and alumni network to find companies that could use a good undergrad math student - these are rarer but they do exist. [Update: mga mentioned the financial sector, which is a great idea too. I was a joint math/CS major, and some of my math friends went straight from college to companies like Fair Isaac (FICO).]

You will need more experience before you can explore certain opportunities - an undergrad math degree doesn't open many doors by itself - but you should try to get that experience working in development, not in IT. Or if you're serious about working on research problems, also consider applying to grad schools in either math or CS.

There are lots of companies that need people who can solve hard (but not always well defined) problems, and being a programmer is certainly a plus. E.g. check out companies that are famous for effectively applying both, e.g. Wal-Mart and Amazon.com (seriously, the latter poaches from the former); prior to the financial industry meltdown, that would have been a place to look into (e.g. many MIT EECS grads used to go there first).

If you're good at problem solving and passable at putting it into code (hey, others can clean that up if need be), you ought to be able to find a niche. Maybe first think about the sorts of problems you like to solve?

Look into quantitative finance. There are plenty of math majors roaming around there, and math people with good programming skills are in high demand.

Recommended introductory reading would be Mark Joshi's 'How to be a quant' PDF (http://www.markjoshi.com/downloads/advice.pdf). It's been around for a little while now, but most of it should still be applicable.

My first job after majoring in math was to work for a software company in hollywood. Have you considered working in something graphics-oriented? It doesn't have to be games, there's a lot of math-related work in the film industry. It will take a little bit of work to break into this field, but it's a really good fit for math majors who have some programming experience.