I am a postdoc in mathematics in the UK. When my current position ends, I plan to apply for positions in the US (among other places). Most of these positions will require some teaching experience, including a teaching recommendation. My current position does not require me to teach, but I have the option of teaching tutorials, teaching graduate courses, and supervising student projects. Is one of these kinds of teaching more in line with the experience that US universities are looking for?
It's important to remember that teaching is not the same as "giving lectures in front of a class". So you (or your letter writers) have some leeway in deciding what exactly the letter should cover.
In practice, lots of people are in your situation. (I was too, when I was looking for faculty positions.) What is important is that one of your letter writers covers how you teach, and this can cover the style in which you give professional presentations, interact with students you mentor, etc. All of these are, when explained by someone who has a lot of experience as a teacher, predictive of how good a teacher you will be in the future. Of course, most potential employers would ideally like to see a discussion in these letters how good a teacher you are in a regular classroom -- since that's one of your primary job duties as a professor -- but the selection committee will be able to extrapolate to that from what they find in a letter.
If you do have the opportunity to teach any regular class, then I would try to do that -- it doesn't matter much in that case whether it's an undergrad or grad course. It just rounds out your resume, and gives you more options for potential letter writers, and more material for letter writers to write about if they choose to sit in on your class once or twice.