I need to take Linear Algebra to progress in my major and the course has Calculus III listed as a prerequisite. I've already taken Calculus III and passed with a C, although I didn't learn the smallest bit of it the entire semester (not a good semester for me). Should I reteach myself Calculus III and hold off on taking Linear Algebra until I do, or would that be a waste of time and should I just go ahead and take the Linear Algebra course? I really don't want to waste time teaching myself something I already took...but if I really need the knowledge, I might.
Go ahead, you can graduate without knowing how to integrate proprely in $n$ dimension...
Anyway usually in Italy Linear Algebra is taken the first Year in both Physics and Mathematics, while multivariable calculus is done the second Year.
Indeed not only it is not a prerequisite to understand the subject, but could very well be quite the contrary being Linear Algebra quite helpful in understanding Calculus III.
Indeed the prerequisites of a standard Linear Algebra course are very few, but there could be more advanced courses of Linear Algebra where the study of linear Algebra is applied to some specific topic (e.g. Differential Equations, Operator Theory, etc...). In this case it may be needed a knowledge of calculus, but usually just the very basics of one or multiple variables. Generally speaking even in this courses the most calculus-involving construction you'll see is the Jacobian.