In this paper, we describe a broad class of problems arising in the context of designing codes for DNA computing. We primarily focus on design considerations pertaining to the phenomena of secondary structure formation in single-stranded DNA molecules and non-selective cross-hybridization. Secondary structure formation refers to the tendency of single-stranded DNA sequences to fold back upon themselves, thus becoming inactive in the computation process, while non-selective cross-hybridization refers to unwanted pairing between DNA sequences involved in the computation process. We use the Nussinov-Jacobson algorithm for secondary structure prediction to identify some design criteria that reduce the possibility of secondary structure formation in a codeword. These design criteria can be formulated in terms of constraints on the number of complementary pair matches between a DNA codeword and some of its shifts. We provide a sampling of simple techniques for enumerating and constructing sets of DNA sequences with properties that inhibit non-selective hybridization and secondary structure formation. Novel constructions of such codes include using cyclic reversible extended Goppa codes, generalized Hadamard matrices, and a binary mapping approach. Cyclic code constructions are particularly useful in light of the fact we prove that the presence of a cyclic structure reduces the complexity of testing DNA codes for secondary structure formation.