I'll add, my "cross browser checking" has become lax in recent years.
I think it's validation [validator.w3.org] that has made me lazy. The reason being is that if I validate every page without fail, including script output, when I do go to check it in various browsers, it's always spot on. There are always "minor" differences - slight bit more space here, type slightly larger there, but if I design in such a way that these differences don't cause navigation rows to run on two lines or otherwise blow up the layout, it's a safe bet it's going to fly well across all browsers.
For a new design, definately, I run it through FF, IE 7, then a machine with IE 6. I rarely worry about Safari, it was one of the first most compliant browsers, but run the design through the Windows version. If it makes it through IE 6, chances are very good it's going to fly. Repeat all tests with zoom and text enlarge/reduce. Then I have an old Mac here through which I test on . . .
get ready for it . . .
Mac IE 5 and Mac NN 4.
When you pick your jaw up off the floor . . . I know these are extinct, and I likely have the only two copies of them in existence, but I consider them my "acid tests." There's little doubt no one is using these, but if it flies in these, I'm feeling good about the implementation.
The ones you really have to worry about are ones that rely on font availability and font sizing to "work." The most common example is a navigation using text-based links within a fixed width container. Fonts will render very differently in, say, Linux browsers versus Windows, even with a basic arial, helvetica, sans-serif. Design in enough wiggle room to stay out of trouble, and you should be OK.