Based on the Baker-Akhiezer function on Riemann surfaces, a general method is proposed to solve spectral problems associated 3 × 3 matrix spectral problems on theta-function periodic backgrounds. Taking the derivative Yajima-Oikawa equation as an example, exact rogue-wave solutions and breather solutions on theta-function periodic backgrounds are constructed. The theta-function periodic solutions are so general that all the Jacobi elliptic function periodic solutions are their special reductions.