  We study the operator space UMD property, introduced by Pisier in the context
of noncommutative vector-valued Lp-spaces. It is unknown whether the property
is independent of p in this setting. We prove that for 1<p,q<\infty, the
Schatten q-classes Sq are OUMDp. The proof relies on properties of the Haagerup
tensor product and complex interpolation. Using ultraproduct techniques, we
extend this result to a large class of noncommutative Lq-spaces. Namely, we
show that if M is a QWEP von Neumann algebra (i.e., a quotient of a C^*-algebra
with Lance's weak expectation property) equipped with a normal, faithful
tracial state \tau, then Lq(M,\tau) is OUMDp for 1<p,q<\infty.
